signature searching in a networked collection of files

Signature Searching in a NetworkedCollection of Files

Zhongwen Ying and Thomas G. Robertazzi, Fellow, IEEE

Abstract—A signature is a data pattern of interest in a large data file or set of large data files. Such signatures that need to befound arise in applications such as DNA sequence analysis, network intrusion detection, biometrics, large scientific experiments,speech recognition and sensor networks. Related to this is string matching. More specifically we envision a problem where long lineardata files (i.e., flat files) contain multiple signatures that are to be found using a multiplicity of processors (parallel processor).This paper evaluates the performance of finding signatures in files residing in the nodes of parallel processors configured as trees,two dimensional meshes and hypercubes. We assume various combinations of sequential and parallel searching. A unique featureof this work is that it is assumed that data is pre-loaded onto processors, as may occur in practice, thus load distribution timeneed not be accounted for. Elegant expressions are found for average signature searching time and speedup, and graphical resultsare provided.

Index Terms—Database search, signatures, tree networks, mesh networks, hypercube networks

Ç

1 INTRODUCTION

A signature is a relatively small data pattern of interestembedded in a very large (in this paper sequential)

data file. It is assumed signatures are temporally distinct anddo not overlap each other. That is, there can be multiplesignatures in a file. Because the files we study are much longerthan the signatures, it is assumed that signatures haveinfinitesimally small length. Such signature searching occursin network security, signal processing, medicine, imageprocessing, and sensor technology and many other fields.

Most previous work on signature searching (known astemplate matching and string matching) develop algorithmfor the detailed matching process. This paper, like [1], [2]addresses an upper level view of signature searchinginvolving system performance evaluation. However wenow briefly summarize some string matching work.

String searching, which is similar to our concept ofsignature searching, is a special case of pattern searching.String searching generically involves finding a pattern oflength m in a text of length n over some alphabet. The worstcase complexity of exact string matching is O(n) but theproportionality constant of the linear term can be verydifferent depending on the string matching algorithm,ranging from m for the naBve algorithm to 2 for the Knuth-Morris-Pratt algorithm [3].

Approximate string matching involves string matchingthat allows errors. That is, the pattern and/or text suffer somecorruption. Applications include noisy channels, speechrecognition, hand writing recognition, finding DNA se-quences in the presence of mutations and text searching.

Approximate string matching algorithms utilize some dis-tance metric to quantify the amount of difference betweentwo strings. For instance, the edit distance is the number ofdifferences between two strings. The computational complex-ity of approximate string matching can range from linear toNP complete depending on the error mechanism [4].

For on-line algorithms, it is assumed that the text isnot known in advance. For off-line algorithms the text canbe pre-processed, thus indexing can be used [5]. On-linealgorithms generally consist of a phase of entering thestring into a data structure and a phase of looking fora match using the data structure [6].

In this paper we assume the data is stored in flat files (i.e.,very long linear sequences of data) stored at nodes of certaininterconnection networks. We assume linear (in the file size)computational complexity which applies to exact stringmatching and some approximate string matching. Naturallymore sophisticated database methodologies are possible andflat file are often converted into other structures [7], [8] but forinitial raw data processing flat files are natural [9].

We envision a scenario where files containing signaturesare placed on a multiplicity of (parallel) processors tiedtogether by an interconnection network [1], [2]. Unlike thework in much of the divisible load theory literature [2],[10], [11], [12], we do not take the time to distribute the loadto the processors and links into account. Rather we assumethe files are pre-loaded onto the processors prior to timet ¼ 0. This is relevant in certain applications. Load is oftenspontaneously distributed to processors without beingscheduled, monitored or timed. Our goal is to determineexpected search time and speedup under a variety ofsearch protocols that largely differ in a number of aspects.These include:

1. Whether the number of signatures in a file at a nodeis known or unknown a priori. The latter is morelikely in general but the former could also occur. Forinstance when a search is done for ‘‘management’’

. The authors are with the Department of Electrical and Computer Engineering,Stony Brook University.

Manuscript received 21 Aug. 2012; revised 27 Aug. 2013; accepted 24 Sept.2013. Date of publication 7 Oct. 2013; date of current version 21 Mar. 2014.Recommended for acceptance by K. Li.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference the Digital Object Identifier below.Digital Object Identifier no. 10.1109/TPDS.2013.258

1045-9219 � 2013 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 25, NO. 5, MAY 2014 1339

in a company database, the number of signaturesmatches may be generally known a priori.

2. Whether a data file stored at a node contains one ormore than one signature. For instance, a database ofcompany employee information may contain onematch for a given individual, but multiple matchesfor individual who are ‘‘management’’.

3. If no signatures are stored in a node, whether it impliesthat there are no signatures stored in the childrennodes of the node. For instance a node may storeaggregated/summarized national data and childrennodes store more detailed state/provincial data. It maybe possible in searching the national nodes to findchildren nodes that should be searched and those thatneed not be searched.

4. Differing degrees of sequentiality and concurrencyin the search strategy. Both have been investigatedin the divisible load scheduling literature over theyears [10], [11], [12].

These are important assumptions that may be made inspecific applications and we try to study this problem in acomplete manner. In this context this work examines tree,mesh and hypercube interconnection networks [13], [14],[15], [16]. We look at trees, meshes and hypercubes mainlybecause these are fundamental interconnection networks.Trees are often used as spanning trees to distribute load inother types of interconnection networks. Meshes are oftenused as interconnection networks in parallel processors.Meshes are particularly well suited to networks on chips[17], [18], [19]. Hypercubes are widely used in parallelprocessors. One could certainly examine other interconnec-tion networks but space limitations prevent this in this paper.

In earlier work the expected search time to find eitherone [1] or multiple signatures [2] with a uniform distribu-tion of signatures in a flat file was found analytically. Thisearlier work involved trees and daisy chains and incorpo-rated, unlike this work, load distribution time to processorsover a network.

The rest of the paper is organized as follows. Section 2discusses searching time and speedup in tree networks fordifferent cases. The searching time and speedup in meshnetworks (store and forward) for different cases are discussedin Section 3. A comparison of the speedups for different typeof networks is presented in Section 4. Finally, this paperconcludes with some possible extensions in Section 5.

2 TREE NETWORKS

A general tree is shown in Fig. 1. Assume that in a tree anode is a structure which contains exactly one file. Eachnode has one, two or more child nodes, which are below itin the tree space. A node that has a child is called the child’sparent node.

A node has at most one parent. Nodes that do not haveany children are called leaf nodes. They are also referred toas terminal nodes. A node’s height is the length of thelongest downward path to a leaf from that node. The root’sheight is the height of the tree. The depth of a node is thelength of the path to its root. The nodes in the same depth ofthe tree are said to be at the same level.

The height of the multiple-level tree in this paper is H.Throughout this paper there is a single file, possibly contain-ing signatures, in each node. In all the cases for tree networkswhich we examine the number of children nodes ni per nodeis a random variable from 1 to N , possibly different for eachlevel. But at the same level the number of children in eachsubtree is the same. In this section if a parent node has asignature(s), its children nodes may also have signatures.If the parent node does not have any signature of interest,its children nodes also have no signatures and do not needto be searched. Some dependency between signature occur-rences is thus being assumed. This may model locality ofreferenceVif a node has a signature(s), there may be relatedinformation (signatures) on its children.

The lengths of all the files in the tree are same, L (bits).The inverse searching speed for one processor is w (s/bit).Suppose that the expected time we need to search a filewith signature is X, because the expected position ofsignature is in the middle of such file (i.e. uniformdistribution in linear file assumption). The time to searchthe whole file is 2X. while 2X ¼ L � w. The purpose of thispaper is to calculate the time we need to find all of the fileswith signatures. We will discuss several cases in Sections 2.1and 2.2 using the assumptions listed in the introduction.

2.1 Number of Signatures is Unknown

2.1.1 Each File Has at Most One SignatureLet Mi be the number of signatures in the ith layer and ni bethe number of children nodes in each subtree in the ithlayer. If there is at most one signature in one file,Mi �Mi�1ni �Mi�2ni�1ni � � � � � n1n2 . . .ni. The expectedtime to search one file is X, the searching process for afile will stop as soon as we find one signature (as also truein Sections 3.1.1 and 3.2.1).

Only one node in the tree commands all the nodes: First,assume that only the top most root node in the multi-leveltree commands all the nodes to search, for every layer,every file in the layer and each node replies to the top nodewhether there is a signature in the node. That is, one layer issearched sequentially at a time. The expected time to searchthe entire tree is

Tu1 ¼ n1X þM1n2X þ � � � þMH�1nHX

¼XHi¼1

Mi�1ni �X: (1)

Here, M0 ¼ 1.

Fig. 1. Multi-level tree network.

IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 25, NO. 5, MAY 20141340

Now we consider a homogeneous case that for all thesubtrees in a certain layer the probability of a signature isthe same: pi for the ith layer. Equation (1) will be Tu1 ¼ n1Xþp1n1n2X þ � � � þ ð

QH�1j¼1 pjnjÞnHX ¼

PHi¼1ð

Qi�1j¼1 pjnjÞniX. A

homogeneous case occurs if p1 ¼ p2 ¼ � � � ¼ pH�1 ¼ p,

Equation (1) will be Tu1 ¼PH

i¼1ðQi

j¼1 njÞpi�1X. The fully

homogeneous case occurs if the number of files in all ofthe subtrees is the same: n, and the probability of signatureis also the same: p, then the expected search time is

Tu1 ¼PH

i¼1 nipi�1X ¼ pHnHþ1�n

pn�1 X.

Every root node of each local tree commands the nodesbelow it: In this case each level is searched sequentially,within each subtree the search is sequential but all subtreesat the same level are searched in parallel. The time to searchthe ith layer is niX. The total expected time to search themulti-level tree is

Tu2 ¼ n1X þ n2X þ � � � þ nHX ¼XHi¼1

niX: (2)

The speedup compared with case 2.1.1.1 is

�u1 ¼PH

i¼1 Mi�1niPHi¼1 ni

: (3)

For the fully homogeneous case, �u1 ¼ pHnHþ1�npn�1

1nH ¼ðpnÞH�1

ðpn�1ÞH .

All the files in every Level are Learched in Parallel, Levelsare Searched Sequentially: In this case each level is searchedsequentially and within a level the nodes are searched inparallel. For every level the search time is only X. Thetotal search time is Tu3 ¼ HX. The speedup compared withcase 2.1.1.1 is

�u2 ¼PH

i¼1 Mi�1niH

: (4)

For the fully homogeneous case, �u2 is ððpnÞH�1Þn

ðpn�1ÞH .

All the files in the tree are searched in parallel: This case isthe fastest search method. All the files in the entire tree are

searched in parallel. The total search time is Tu4 ¼ X andthe speedup is

�u3 ¼XHi¼1

Mi�1ni: (5)

For the fully homogeneous case, �u3 is ððpnÞH�1Þn

ðpn�1Þ .

It is apparent that as the height increases, all the threespeedups also increase, and �u1 G �u2 G �u3, �u3 is muchlarger than �u1 and �u2 when the height is high. From

equations (3), (4), (5) we know �u3 ¼ H � �u2, �u2 ¼PH

i¼1ni

H �u1,

the expected mean number of

PH

i¼1ni

H isN2H

H ¼ N2 . This

confirms that the highest performing strategies, if it canbe implemented, is to search all files in parallel.

2.1.2 Multiple Signatures Can Exist in One FileOne should search the whole file to find all of thesignatures in each file. If there are multiple signatures inone file, the expected search time is 2X (Sections 3.1.2and 3.2.2), Mi can be larger than Mi�1 � ni, and we assumethat those Mi signatures exist in Ni files (each of Ni fileswith at least one signature).

Only one node in the tree commands all the nodes: Theexpected search time in layer i is Ni�1ni � 2X, for Ni fileswith signatures, Ni �Mi. The total expected search time is:

�Tu1 ¼XHi¼1

Ni�1ni � 2X: (6)

Every root node of each local tree commands the nodesbelow it: Again the levels are searched sequentially, withineach subtree the search is sequential but all subtrees at thesame level are searched in parallel. The expected searchtime in layer i is ni � 2X. The total expected search time is:

�Tu2 ¼XHi¼1

ni � 2X: (7)

The speedup is:

��u1 ¼PH

i¼1 Ni�1ni � 2XPHi¼1 ni � 2X

¼PH

i¼1 Ni�1niPHi¼1 ni

: (8)

All the files in every level are searched in parallel, levels aresearched sequentially: The same case as before, �Tu3 ¼ H � 2X.

The speedup is ��u2 ¼PH

i¼1Ni�1ni

H .

All the files in the tree are searched in parallel: The samecase as before, �Tu4 ¼ 2X. The speedup is ��u3 ¼

PHi¼1 Ni�1ni:

The speedup is plotted in Fig. 2. As a baseline result, it isassumed that the distribution of signatures in every file is a

Poisson distribution: P ðm ¼ kÞ ¼ �k

k! e�� (m is the number of

signatures in every file). We can see from Fig. 3 that when �goes up from 1 to 20, the speedup ��u1 increases quickly firstand then saturates. That is because initially the number ofsubtrees needed to be searched increased and later onewinds up searching all, but not more than all, files in alevel, then the speedup saturates.

2.2 Number of Signatures is KnownWhen the number of signatures is known, the searchingprocess will stop as soon as the last signature is found,

Fig. 2. Speedup when the number of signatures is unknown and eachfile can have multiple signatures.

YING AND ROBERTAZZI: SIGNATURE SEARCHING IN A NETWORKED COLLECTION OF FILES 1341

thus the time should be shorter than that of the case where thenumber of signatures is unknown. Two cases are discussed.

2.2.1 Each File Has at Most One SignatureIn this case assume that there are m signatures in n files in asingle level of the tree. Each file can have at most onesignature. Each node holds one file. We focus on the lastsignature because the searching process will not stop until thelast signature is found, thus the search time depends on theposition of the last signature. Because there are m signaturesand every file can have at most one signature, the lastsignature should be in themth; ðmþ 1Þth; . . . ; nth file. If the lastsignature is in the ðm� 1Þth or even a file to the left ofmth file,there will be multiple signatures in some files, which is acontradiction. The probability that the last signature is in themth file is (there are m� 1 signatures in m� 1 files)

Pm ¼m�1m�1

� �nm

� � : (9)

And the expected search time is

tm ¼ PmðmXÞ: (10)

The probability that the last signature is in the ith file is(there are m� 1 signatures in i� 1 files) for i � m� 1 is

Pi ¼i�1m�1

� �nm

� � : (11)

And the expected search time is

ti ¼ Pi � mX þ ði�mÞ2X� �

¼i�1m�1

� �nm

� � ð2i�mÞX: (12)

Thus, the expected search time Ts for this single level is

Tn;m ¼Xni¼m

ti ¼Xni¼m

i�1m�1

� �nm

� � ð2i�mÞX: (13)

We plot the expected search time as the number ofsignatures increases. In Fig. 3 the number of files n ¼ 100,

and we can conclude that when m ¼ 13, the expectedsearch time is the largest.

We derive the expression for the maximum value of thefunction of Fig. 3 in the supplementary file which isavailable in the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/258.

d

dmTm;n ¼

2n�m2 � 2mþ 1

ðmþ 1Þ2�X ¼ 0 (14)

Then we obtain m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nþ 2p

� 1. If n ¼ 100, then

equation (14) will be ddmTm;n ¼ 200�m2�2mþ1

ðmþ1Þ2 , let ddmTm;n ¼ 0,

then we get the maximum expected search time occur whenm ¼ 13:21 which is close tom ¼ 13. For this graph if a file hasa signature the expected search time is X, and if there is nosignature, it is 2X. Initially the mean time to find all signaturesincreases as more signatures are located closer to the last file.Eventually though as the number of signatures increases mostfiles have a signature forcing the average search time per filecloser to X rather than 2X and the curve decreases.

Up to this point in this section the expected time for onesingle level has been solved. Next the expected time tosearch anH level tree is considered. There are four cases justas in the case where the number of signatures is unknown.

Only one node in the tree commands all the nodes: Assumethat there aremi;j signatures in the jth subtree in layer i, andMi ¼

PMi�1

j¼1 mi;j, Mi stands for the total number of sig-natures in the ith layer. Then for the jth subtree in layer i, the

expected search time Tni;mi;j¼Pni

i¼mi;j

i�1mi;j�1

� �nimi;j

� � ð2i�mi;jÞX.

The total expected search time is thus:

T1 ¼XHi¼1

XMi�1

j¼1

Tni;mi;j: (15)

Assume m1;1 ¼ m1 ¼M1 and M0 ¼ 1.

Every root node at local tree to command the nodes below it:Again, the levels are searched sequentially, within each subtreethe search is sequential. However all subtrees at the same levelare searched in parallel. For a given layer, the search time inthis layer is the longest expected search time in the layer’ssubtrees. Assume that in the ith layer the largest expectedsearch time is max½Tni;mi;j

� (j is from 1 to Mi�1 while Mi hasbeen defined before), The total time to search the entire tree is:

T2 ¼XHi¼1

max Tni;mi;j

� �: (16)

The speedup is:

�1 ¼PH

i¼1

PMi�1

j¼1 Tni;mi;jPHi¼1 max Tni;mi;j

� � : (17)

For the fully homogeneous case where n1 ¼ n2 ¼ . . . ¼nH ¼ n and m1 ¼ m2;1 ¼ . . . ¼ mH;1 ¼ . . . ¼ m, the numberof signatures in every subtree is the same in every layer inthe first case. Then

�1 ¼PH

i¼1 mi�1Tn;m

H � Tn;m¼ mH � 1

Hðm� 1Þ : (18)

Fig. 3. Single level tree when the number of signatures is known.


All the files in every level are searched in parallel, levels aresearched sequentially: Discussed as before, T3 ¼ HX and thespeed up compared with case 2.2.2.1 is

�2 ¼PH

i¼1

PMi�1

j¼1 Tni;mi;j

HX: (19)

For the fully homogeneous case thatn1 ¼ n2 ¼ . . . ¼ nH ¼ nand m1 ¼ m2;1 ¼ . . . ¼ mH;1 ¼ . . . ¼ m, �2 ¼ mH�1

Hðm�1ÞTn;m ¼mðmH�1Þð2n�mþ1Þ

Hðm2�1Þ .

All the files in the tree are searched in parallel: Discussed

as before T4 ¼ X, and the speedup �3 ¼PH

i¼1

PMi�1

j¼1Tni;mi;j

X. For

the fully homogeneous case that n1 ¼ n2 ¼ � � � ¼ nH ¼ nand m1 ¼ m2;1 ¼ � � � ¼ mH;1 ¼ � � � ¼ m, then �3 ¼ mH�1

Hðm�1ÞTn;m ¼ mðmH�1Þð2n�mþ1Þ

m2�1 .

This is apparent that �3 9 �2 9 �1, because for the fully

homogeneous case, �3 ¼ H�2, �2 ¼ mð2n�mþ1Þmþ1 �1, absolutely

mð2n�mþ1Þmþ1 9 1.

2.2.2 Multiple Signatures Can Exist in One FileFirst we consider a single level tree which has n files and msignatures. We combine all the files logically in one levelinto a single file. We also assume that the capacity of onefile is L bits, so that the length of the combined file is n � L.The position of the signatures is X1; X2; . . . ; Xm, ðnL �X1; X2; . . . ; Xm � 0Þ, X � Uð0; nLÞ.

P ðXmax ¼ xÞ ¼m

1

� �� 1

nL� x

nL

� �m�1: (20)

The expected length of the last signature position:

Xmax ¼ZnL

0

m

nL� x

nL

� �m�1�xdx ¼ m

mþ 1� nL: (21)

The search inverse speed is w s/bit. Then the expectedtime we need to search one single level is �Tn;m ¼ Xmax�

w ¼ mmþ1 � n � L � w, here L � w ¼ 2X, thus

�Tn;m ¼2m

mþ 1nX: (22)

This is plotted in Fig. 4. Here the number of files is 10 andthe maximum number of signatures is 50. From the figure wecan see that the expected search time will increase and saturateas the number of signatures increase. That can be seen fromEquation (22), as m increases, �Tn;m will be close to 2nX.

Until now in this section we have solved the expectedtime in one single level, now we consider the expected timewe need to search the H level tree. Assume that the mi

signatures are in �ni files. Apparently �ni � mi. The fourcases are discussed as below.

Only one node in the tree commands all the nodes: For theith layer, there are Ni�1 files that should be searched, thereare mi;1; mi;2; . . . ; mi;Ni�1

signatures in �ni;1; �ni;2; . . . ; �ni;Ni�1

files, Ni ¼PNi�1

j¼1 �ni;j. The total time is

�T 1 ¼XHi¼1

XNi�1

j¼1

�Tni;mi;j: (23)

Note that N0 ¼ 1 and �ni;j � mi;j.

Every root node at local tree to command the nodes below it:In this case the search time in the ith layer is the time to searchone of the subtrees which has the largest number of signatures(like the case discussed before). The total search time is:

�T 2 ¼XHi¼1

max �Tni;mi;j

h i: (24)

Note that j is from 1 to Ni�1. The speed up is:

��1 ¼PH

i¼1

PNi�1

j¼1�Tni;mi;jPH

i¼1 max �Tni;mi;j

h i : (25)

All the files in every level are searched in parallel, levels aresearched sequentially: Discussed as before, �T 3 ¼ 2HX. Thespeedup compared with case 2.2.2.1 is

��2 ¼PH

i¼1

PNi�1

j¼1�Tni;mi;j

2HX: (26)

All the files in the tree are searched in parallel: The totalsearch time in this case is 2X. The speedup here is:

��3 ¼PH

i¼1

PNi�1

j¼1�Tni;mi;j

2X: (27)

The speedup figure for these cases has been plotted inFig. 5. From the figure we can see that ��3 9 ��2 9 ��1. Thereason is similar to the case in 2.2.1.

3 MESH NETWORKS: STORE AND FORWARD

A regular two dimensional mesh network of processors[11] is shown in Fig. 6. It is a commonly used interconnec-tion network. In this network structure each processor islocated in the corners of four rectangles and has fourneighbors. The central processor is called the originator, itcan communicate information or transport data to its four

Fig. 4. The expected search time when the number of signatures isknown and multiple signatures can exist in one file for a single level tree.


neighboring nodes. As in trees, the central node is assumedto be layer 0, and its four neighboring nodes in layer 1, thenodes which are neighbors to the nodes in layer 1 but not inlayer 0 are in layer 2, . . ., the nodes which are neighbors tothe nodes in layer i but not in layer i� 1 are in layer iþ 1, asshown in the mesh structure. A node can only sendmessages or data to its neighboring nodes. If the node inthe north of layer 2 wants to send a message to the node inlayer 0, first it should send message to the node in the northtop of layer 1, then the message will be transferred to layer 2.

Assume that there are N layers (as shown in Fig. 6) andthere is at most one signature in every node in a mesh network.There are 4i nodes in the ith layer. Different from the treenetwork, the case that every node can have a signature will bediscussed in Section 3.1. That is, if a node has no signature, itschildren nodes can still have signatures. Assume the averagetime to search every node in the mesh network is the same:X, and again there is at most one signature in one file.

3.1 Every File Can Have Its Own SignatureRegardless of the Upper Layer Node

3.1.1 The Number of Signatures Is UnknownIn this case every node should be searched. There arethree cases.

Only one node in the network commands all the nodes: Thecentral node which is in layer 0 commands each node tosearch for signatures. As soon as the central node sends amessage to command one node to start to search its file, thecentral node will wait until it receives a message indicatingwhether there is any signature in the file. All of the nodes inthe mesh network are searched one by one. It is apparentthat the searching time is the longest, the searching time inthe ith layer is 4i �X. The total expected search time is:

T1 ¼ 4X þ 8X þ . . .þ 4NX ¼ 2NðN þ 1ÞX: (28)

Every root node of each local network commands the nodebelow it: The searching time in layer one is X. For layer two,there are 8 nodes to be searched. We assign two nodes inlayer two to be searched by each node in layer one, the

search time should be 2X. For layer three, there are twocases, the first case is for the four nodes in the upmost, ordownmost, or leftmost, or rightmost direction. The ex-pected search time for these four nodes is 3X, because thereare three neighboring nodes in their lower layer; and for thesecond case (the remaining nodes), there are only twoneighboring nodes in their lower layer. Thus the expectedsearch time is 2X. However, we can eliminate X searchingtime for both cases, because every node in layer three exceptthe four top nodes in four directions have two upper nodes(different from the tree network). If one node for the node inlayer three is commanded by one node in layer two, the othernode in layer two need not command it to search for asignature. For layer three, the searching time is 2X, the samefor layer four, five, . . .. The total expected search time isT2¼ 4Xþ2Xþ2Xþ� � �þ2X¼ 4Xþ2ðN�1ÞX¼ ð2Nþ2ÞX.

All the files in every layer are searched in parallel, layers aresearched sequentially: Like the third case in the tree network,the search time in every layer isX. The total time is T3 ¼ NX.

All the files in the mesh network are searched in parallel:The expected search time in this case is only T4 ¼ X.

Then we calculate the speedup in these four cases: �1¼T1

T2¼N , �2¼T1

T3¼2ðN þ 1Þ, �3 ¼ T1

T4¼ 2NðN þ 1Þ, and the speed-

up figure has been plotted in Fig. 7. From the figure we knowthat �3 9 �2 9 �1, that is because �2

�1¼ 2ðNþ1Þ

N 9 2 and �3

�2¼ N .

3.1.2 The Number of Signatures Is KnownOnly one node in the network commands all the nodes: This

case is mostly similar to the case in the tree network: assumethere are mi signatures in the ith layer, we focus on the lastsignature. There are 4i files in the ith layer, the probability thatthe last signature is in the jth file in the ith layer is:

Pj ¼j�1mi�1

� �4imi

� � : (29)

Fig. 5. Speedup when there can be multi-signatures in one file and thenumber of signatures in a level is known.

Fig. 6. Mesh network.


The expected searching time in the ith layer is:

�T i ¼X4ij¼mi

j�1mi�1

� �4imi

� � ð2j�miÞX: (30)

The total time to search the mesh network is �T 1 ¼PNi¼1

�T i. The time should be less than T1.

Every root node of each local network commands the nodesbelow it: For layer one, the expected search time is the same

as T2:1 ¼P4

j¼m1

j�1m1�1

� �4m1

� � ð2j�m1ÞX. For layer two, three and

the next layers, there are two types of nodes. The first typeof nodes is those nodes which have three children nodes intheir lower layer, from Fig. 2 the first type of nodes are inthe top most position of east, west, south, north fourdirections. The other type of nodes are the other nodes,which only have two children nodes in their lower layers.However, the searching time for the two types nodes arenot 2X and 3X, because every node except the four topnodes have two parent nodes. Assume that in the first Xsearching time all the nodes except the four top nodes aresearched, and for the parent node only one nodecommands one node. If all the signatures are found, thesearching time in this layer is X. If there are somesignatures in the top four nodes, we need to add X moretime to find the signature, then the searching time shouldbe 2X. The probability that there are no signatures in thetop four nodes in the ith layer is:

P2:i ¼4i�4mi

� �4imi

� � : (31)

The total time of the mesh network in this case is:

�T 2 ¼ �T 1 þXNi¼2

P2;iX þ ð1� P2;iÞ2X� �

: (32)

All the files in every layer are searched in parallel, layers aresearched sequentially: The same as the last case we havediscussed, the total time is �T 3 ¼ NX.

All the files in the mesh network are searched in parallel:The total time is �T 4 ¼ X.

Now we calculate the speedup for these four cases.

��1 ¼�T 1�T 2¼

PN

i¼1�T i

�T 1þPN

i¼2P2;iXþð1�P2;iÞ2X

, while �T i and P2;i have been

defined before (see Equation(30) and (31)). ��2 ¼�T 1�T 3¼PN

i¼1�T i

NX,

��3 ¼�T 1�T 4¼PN

i¼1�Ti

X. The speedup figure has been plotted in

Fig. 8. From the figure we can know that ��3 9 ��2 9 ��1. That

is because ��3 ¼ N � ��2, ��2

��1¼

�T 1þPN

i¼2P2;iXþð1�P2;iÞ2XNX

9 1.

3.2 Only the Nodes Whose Parent Node Has aSignature Can Have a Signature

In this case those nodes whose parent node do not havea signature, will also have no signatures. Only thosenodes whose parent node has a signature need to besearched. This case is so complex that one can not find ageneral equation to solve it. We simulated 100,000 meshnetworks with different probabilities of finding signa-tures in the files, and calculated the search times. Inthis section we only discuss the case that the number ofsignatures is unknown.

3.2.1 Only One Node in the Network CommandsAll the Nodes

The mesh network structure is divided into four parts: up-left, up-right, down-left and down-right. The nodes inthese four parts will search their children nodes in fourdifferent directions, which is also up-left, up-right, down-left and down-right. It is probable that those nodes, whoseparent node has a signature, will have signatures. Thenodes that are not in the top position of every layer havetwo parent nodes, the case that either of them or both ofthem have a signature will lead the child node to possiblyhave a signature. We can get the mesh structure by creatingthe signature distribution layer by layer. The next step is to

Fig. 7. Speedup when the number of signatures is unknown in a meshnetwork and each node can have at most one signature.

Fig. 8. Speedup when the number of signatures is known in a meshnetwork and there is at most one signature in each node.


calculate the search time in every layer, which depends onthe signature distribution in the upper layer. The four topnodes have three children nodes, and the other nodes havetwo children nodes. Thus the total search time for a certainlayer is the number of top nodes, which have a signature tobe searched, times 3X, plus the number of the other nodes,which have a signature, times 2X. For this algorithm Fig. 9shows the search time for every layer as the probability ofsignatures increases. Smaller values of p result in thesignature searching processing stopping earlier as irregularsignature free boundaries form around the mesh networkthat prevent further signature searching.

Fig. 9 shows that when the probability of finding asignature in every node is higher, the expected time tosearch every layer will be longer. When the probability of asignature is exactly 1, that means every node will have asignature, the time to search the ith layer is 4i. When theprobability of a signature is equal or less than 0.5, thenumber of nodes which have signatures will decrease asthe layers increase until no node will have signatures in acertain layer. When the probability of signature is equal to0.6, there are almost always nodes which have signaturesand the number of nodes which have signatures in everylevel remains almost constant. When the probability ofsignature is equal or more than 0.7, the number of nodeswhich have signatures increases.

3.2.2 Every Root Node of Each Local NetworkCommands the Nodes Below it

For the four top nodes in the east, west, north, and southdirections, if both of the two neighboring nodes in theirlayer have no signatures (i.e they will not command theirlower layer node to search their files), the searching timefor the top nodes is 3X. If one of the neighboring nodes orboth of them have signatures, we can assign the nodewhose parent nodes are the top node and its neighboringnode to be commanded by the node neighboring the topnode. Thus the searching time will be cut to 2X. For theother node, the maximum searching time is 2X, when these

nodes have signatures. All the local mesh networks (thenode and its children nodes) search their own nodes inparallel. The searching time for a certain layer depends onthe signature distribution of the top node of the upperlayer, as mentioned before, it will be 3X, 2X, or 0 (thereare no signatures in the whole parent nodes). We simulatethis case in Fig. 10.

From Fig. 10, when the probability of a signature is equalor less than 0.6, the searching time is shorter and shorteruntil we need not search the signature in the lower layer asthe layer is larger. That’s because smaller values of p resultin the signature searching processing stopping earlier asirregular signature free boundaries form around the meshnetwork that prevent further signature searching. Whenthe probability of signature is equal or higher than 0.7, forthose higher layer the searching time is exactly 2X. That isbecause the probability that the top node has a signatureand meanwhile the two neighboring nodes have nosignatures is almost impossible. Thus the searching timefor those high layers is exactly 2X.

The speedup for case 3.2.1 and case 3.2.2 (speedup 1)has been depicted in Fig. 11. From it we can conclude thatwhen the probability of a signature is below 0.6, the speedupsare saturated at less than 4. We can not reduce much searchingtime by using the method in case 3.2.2 when the probability ofa signature is below 0.6 as the number of layers increases. Thatis because when the probability of a signature is below 0.6, thesearching time for every layer in case 3.2.1 will decrease as thenumber of layers increases (see Fig. 9). While the probability ofa signature is more than 0.6, the speedup rises much faster.That is because in Fig. 9 we know the the searching time willgo up quickly, while the searching time will be 2X in Fig. 10.

3.2.3 All the Files in Every Layer Are Searched inParallel, Layers Are Searched Sequentially

The searching time for every layer is X. The total searchtime is thus NX. The speedup for 3.2.1 and case 3.2.3(speedup 2) has been depicted in Fig. 12. It can be seen fromFig. 12 the speedup is larger than the speedup in Fig. 11

Fig. 9. Searching time for the mesh network in case 3.2.1 as signatureprobability is varied.

Fig. 10. Searching time for the mesh network in case 3.2.2 as signatureprobability is varied.


when the probability of signature is larger than 0.6, for thesearching time for every layer is the same X. When theprobability of signature is lower than 0.6, the same asFig. 11, the speedup does not increase when the number oflayers increases.

3.2.4 All the Files in the Mesh Network Are Searchedin Parallel

The expected search time in this case is only X. Thespeedup (speedup 3) for case 3.2.4 and case 3.2.1 has beenplotted in Fig. 13. It can be seen that the speedup increasesmuch faster as the number of layers increase, than thespeedup in Fig. 12 when the probability of a signature islarger than 0.6, and when the probability of a signature islower than 0.6, the speedup does not increase as thenumber of layers increase.

4 SEARCHING TIME COMPARISON FOR DIFFERENTTYPES OF NETWORK

The four types of networks considered are tree networks,mesh networks: store and forward, mesh networks: circuitswitched and wormhole routing (discussed in the supple-mentary file available online) and hypercubes (discussed insupplementary file available online). Each of these hasadvantages and disadvantages. Now we want to comparetheir searching time if the total number of nodes is fixed:N .For the tree network, assume that the number of nodes inevery subtree, n, is the same, which is a homogeneous case.The height of the tree should be: 1þ nþ n2 þ . . . þnH�1 ¼ N , then H ¼ lognððn� 1ÞN þ 1Þ. The total searchingtime for case 3.1.1.2 is T1 ¼ nHX ¼ n lognððn� 1ÞN þ 1ÞX.The searching time is plotted in Fig. 14 (here N ¼ 100) forthese four types of network. We can see that when thenumber of nodes in every subtree increases, the searchingtime also increases. For the store-and-forward mesh

Fig. 11. Speedup 1 when the number of signatures is unknown and onlythe nodes whose parent node has a signature can have a signature inthe mesh network.

Fig. 12. Speedup 2 when the number of signatures is unknown and onlythe nodes whose parent node has a signature can have a signature inthe mesh network.

Fig. 13. Speedup 3 when the number of signature is unknown and onlythe nodes whose parent node has a signature can have a signature inmesh network.

Fig. 14. Four types of networks comparison.


network, the number of layers should be: 1þ 4þ � � � þ4ðH � 1Þ ¼ N , then H ¼ 1þ

ffiffiffiffiffiffiffiffiffi2N�1p

2 . The searching time shouldbeT2 ¼ ð2H þ 2ÞX ¼ ð3þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2N � 1p

ÞX. For the circuit switch-ing and wormhole routing mesh network, the searching timeis T3 ¼ logpþ1 NX. For the hypercube network, the searchingtime is T4 ¼ log2NX. In Fig. 14, we can see that the searchingtime for circuit switching and the wormhole routing meshnetwork is short, but this type network is hard to realize. Thesearching time for hypercubes is also very short, and this typecan be easily realized in low dimensionality. For the treenetwork and store-and-forward network, which are easy torealize, if the number of nodes in every subtree is more than 4,the searching time for the store and forward mesh network isshorter, for the tree network when the number of nodes inevery subtree is less than or equal to 4, the searching time forthe tree network is shorter.

5 CONCLUSION

It has been demonstrated that the expected search time forsignatures in a wide variety of search scenarios for tree,mesh and hypercube networks, where load distributiontime is not considered, can be calculated either analyticallyor through simulation. This should also be possible forother types of interconnection networks. Future researchshould consider other types of file structures or statisticalassumptions. This work is of interest in a wide variety ofapplied areas involving signature searching.

ACKNOWLEDGMENT

The authors acknowledge the support of DOE grantDE-SC0003361.

REFERENCES

[1] K. Ko and T. Robertazzi, ‘‘Signature Search Time Evaluation inFlat File Databases,’’ IEEE Trans. Aerosp. Electron. Syst., vol. 44,no. 2, pp. 493-502, Apr. 2008.

[2] Y. Kyong and T.G. Robertazzi, ‘‘Greedy Signature Processingwith Arbitrary Location Distributions: A Divisible Load,’’ IEEETrans. Aerosp. Electron. Syst., vol. 48, no. 4, pp. 3027-3041, Oct. 2012.

[3] R. Baeza-Yates, ‘‘Algorithms for String Searching,’’ AcM sIGIRForum, vol. 23, no. 3/4, pp. 34-58, Apr. 1989.

[4] G. Navarro, ‘‘A Guided Tour to Approximate String Matching,’’ACM Comput. Surveys, vol. 33, no. 1, pp. 31-88, Mar. 2001.

[5] P. Michailidis and K. Margaritis, ‘‘String Matching Algorithms:Survey and Experimental Results,’’ Int’l J. Comput. Math., vol. 76,no. 4, pp. 411-434, Jan. 2001.

[6] Z. Galil and R. Giancarlo, ‘‘Data Structures and Algorithms forApproximate String Matching,’’ J. Complex., vol. 4, no. 1, pp. 33-72,Mar. 1988.

[7] H. Kitakami, T. Shin-I, K. Ikeo, Y. Ugawa, N. Saitou, T. Gojobori,and Y. Tateno, ‘‘Yamato and Asuka: DNA Database ManagementSystem,’’ in Proc. 28th Hawaii Int’l Conf. Syst. Sci., 1995, vol. 5, pp. 72-80.

[8] M. Hoffman and D. Carver, ‘‘Reverse Engineering Data Require-ments,’’ in Proc. IEEE Aerosp. Appl. Conf., 1996, vol. 2, pp. 269-277.

[9] M. Lubeck, D. Geppert, and K. Nienartowicz, ‘‘An Overview of aLarge-Scale Data Migration,’’ in Proc. 20th IEEE 11th NASAGoddard Conf. MSST, 2003, pp. 49-55.

[10] T. Robertazzi, Networks and Grids: Technology and Theory. New York,NY, USA: Springer-Verlag, 2010.

[11] M. Drozdowski, Scheduling for Parallel Processing. New York,NY, USA: Springer-Verlag, 2009.

[12] H. Casanova, A. Legrand, and Y. Robert, Parallel Algorithms. BocaRaton, FL, USA: CRC Press, 2009.

[13] W. Gaazek, ‘‘A Multistage Load Distribution Strategy for Three-Dimensional Meshes,’’ Cluster Comput., vol. 6, no. 1, pp. 31-39,Jan. 2003.

[14] M. Drozdowski and W. Glazek, ‘‘Scheduling Divisible Loads ina Three-Dimensional Mesh of Processors,’’ Parallel Comput.,vol. 25, no. 4, pp. 381-404, Apr. 1999.

[15] K. Li, ‘‘Speed-Up of Parallel Processing of Divisible Loads onk-Dimensional Meshes and Tori,’’ Comput. J., vol. 46, no. 6,pp. 625-631, 2003.

[16] J. Baazewicz and M. Drozdowski, ‘‘Performance Limits of Two-Dimensional Network of Load-Sharing Processors,’’ Found.Comput. Decision Sci., vol. 21, no. 1, pp. 3-15, 1996.

[17] T. Bjerregaard and S. Mahadevan, ‘‘A Survey of Research andPractices of Network-On-Chip,’’ ACM Comput. Surveys, vol. 38,no. 1, p. 1, 2006.

[18] E. Salminen, T. Kangas, V. Lahtinen, J. Riihimaki, K. Kuusilinna,and T. Hamalainen, ‘‘Benchmarking Mesh and Hierarchical BusNetworks in System-On-Chip Context,’’ J. Syst. Architect., vol. 53,no. 8, pp. 477-488, Aug. 2007.

[19] S. Stuijk, T. Basten, M. Geilen, A. Ghamarian, and B. Theelen,‘‘Resource-Efficient Routing and Scheduling of Time-ConstrainedStreaming Communication on Networks-on-Chip,’’ J. Syst. Architect.,vol. 54, no. 3/4, pp. 411-426, Mar. 2008.

Zhongwen Ying received the BS degree inelectrical engineering from Zhejiang University,Hangzhou, China, in 2009. He is currently pursuingthe PhD degree in Stony Brook University since2009. His research interests are in the area ofdistributed and parallel computing and algorithmdesign and he is currently working on signaturesearching algorithm and divisible load theory.

Thomas G. Robertazzi received the BEE from theCooper Union, New York, NY, in 1977 and the PhDdegree from Princeton University, Princeton, NJ, in1981.He is presently a Professor in theDepartmentof Electrical and Computer Engineering at StonyBrook University, Stony Brook NY. In supervising avery active research group, he has publishedextensively in the areas of parallel processing andgrid scheduling, telecommunications network plan-ning, ATM switching, queueing and Petri networks.Prof. Robertazzi has also authored, co-authored or

edited six books in the areas of networking, performance evaluation,scheduling and network planning. For eleven years Prof. Robertazzi wasthe faculty director of the Stony Brook Living Learning Center in Scienceand Engineering. For the past five years he has co-chaired theStony BrookUniversity Senate Research Committee. He is a Fellow of the IEEE.

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.


signature searching in a networked collection of files

Education