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Data Structures

CSCI 132, Spring 2014Lecture 32

Tables I

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Table Lookup

•You can speed up search if you store items in a table.

•Assign 1 item per key, for integer keys ranging from 1 to n.

•To find an item, simply access the index given by its key.

0 1 2 3 4 5 6 7 8

no it by if at we in of an

•Table lookup has running time of O(1)

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Example: Color Tables

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Rectangular Tables

c s c if u n !v e r y

(0, 0)

(2, 3)

Tables are very often in rectangular form with rows and columns.

Most programming languages accommodate rectangular tables as 2-D arrays.

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Rectangular Tables in Memory

c s c if u n !v e r y

(0, 0)

(2, 3)

Row-major order:

c s c i f u n ! v e r y

Column-major order:

c f v s u e c n r i ! y

row 0 row 1 row 2

col 0 col 1 col 2 col 3

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Computing the index in memory

The compiler must be able to convert the index (i, j) of a rectangular table to the correct position in the sequential array in memory.

For an m x n array (m rows, n columns):Each row is indexed from 0 to m-1Each column is indexed from 0 to n - 1

c . . . i f . . . ! v . . . y

row 0 row 1 row 2

Item at (i, j) is at sequential position i * n + j

0 n-1 n 2n2n-1

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Access Arrays

•Multiplication can be slow. To speed things up, we can compute the beginning index of each row and store it in an auxiliary array, called an access array.

0 n 2n 3n ...

0 1 2 3 ...

•To find the position of (i, j), take the value at index i from the access array and add j. (access_array[i] + j)

•In general, an access array is an auxiliary array used to find data that is stored elsewhere.

access_array

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Triangular TablesSome rectangular tables have positions where the entry is always zero.

To save memory, we omit these positions from our table.

Definitions: A matrix is a table of numbers.A lower triangular matrix is a matrix in which non zero entries are only in positions for which i >= j:

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Leaving out the zeros

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Calculating Access Array values

Number of cells preceding the start of row i:0 + 1 + 2 + .... + i = i (i + 1) /2

Each entry (i, j) is located at position:j + i (i + 1)/2

One can compute the values of the access array once and then not again for the entire program.

You do not need multiplication to compute these values:0, 0 + 1, 1 + 2, (1 + 2) + 3, ....

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Jagged Tables

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Inverted Tables

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Functions and Tables•A function starts with an argument and computes a value.•A table starts with an index and looks up a value.

•A function assigns to each element in set A (the domain) a single element from set B (the codomain).•A table assigns to each index in set A (the index set) a single value from set B (the base type).

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The Table ADTA table with index set I and base type T is a function from I into T together with the following operations.

Standard operations:1. Table access: Evaluate the function at any index in I.2. Table assignment: Modify the function by changing its value at a specified index in I to the new value specified in the assignment.

Additional operations:3. Creation: Set up a new function from I to T .4. Clearing: Remove all elements from the index set I , so the remaining domain is empty.5. Insertion: Adjoin a new element x to the index set I and define a corresponding value of the function at x.6. Deletion: Delete an element x from the index set I and restrict the function to the resulting smaller domain.

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Lists vs. Tables

Lists are inherently sequential; Tables are not.

List retrieval requires search O(lgn); Tables do not: O(1)

Traversal is natural for a list, not for a table.