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Searching in a Graph

Jeff EdmondsYork University COSC 3101LectureLecture 55

Generic SearchBreadth First SearchDijkstra's Shortest Paths AlgorithmDepth First SearchLinear Order

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Graph

a

c

b Node ~ city or computer

Edge ~ road or network

Undirected or Directed

A surprisingly large number of problems in computer science can be expressed as a graph theory problem.

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Graph Search

Specification: Reachability-from-single-source s•<preCond>: The input is a graph G (either directed or undirected) and a source node s. •<postCond>: Output all the nodes u that are reachable by a path in G from s.

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Graph Search

Basic Steps:

su

Suppose you know that u is reachable from s

v

& there is an edge from u to v

You know that v is reachable from s

Build up a set of reachable nodes.

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Graph Search

s

reachable

fuj reachable

dvw reachable

ehd reachable

btd

vw

ehd

bt

•How do we keep track of all of this information?

•How do we avoid cycling?

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Graph Search

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Graph Search

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Graph Search

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

i.e. find its neighborsDon’t re-find a node.

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Handle some foundNotHandled node

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

Handle some foundNotHandled nodei.e. find its neighbors

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

# of found nodes.measureprogress

# of handled nodes.Might not increase.

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

Node s is foundNotHandledOther nodes notFound

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

All nodes found.Exit When can’t make any more progress.

No. Might not find all.

When all found nodes are have been handled.Handle some foundNotHandled node

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Graph Search

We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

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lExit All found nodes are handled.

Exit

•<postCond>: Output all the nodes u that are reachable by a path in G from s.

Output Found nodes

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

Exit

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lAll found nodes are handled.

Exit Found nodes are reachable from s. Reachable nodes have been found.

•<postCond>:

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We know found nodes are reachable from s because we have traced out a path.

If a node has been handled, then all of its neighbors have been found.

Graph Search

Exit

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lAll found nodes are handled.

Exit Reachable nodes have been Found. •<postCond>:

Found = handled

handled

notfound

[A B] = [B A]notFound nodes not reachable.

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Graph Search

Specification of Reachability-from-single-source s•<preCond>: The input is a graph G (either directed or undirected) and a source node s. •<postCond>: Output all the nodes u that are reachable by a path in G from s.

EndingInitial ConditionsMake Progress

Maintain Loop InvDefine Exit ConditionDefine Step

Define Measure of Progress

Define Loop Invariants

Define Problem

km

79 km

to school

Exit

Exit

79 km 75 km

Exit

Exit

0 km Exit

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Graph Search

# of handled nodes.

Handle some foundNotHandled node

Time = O(n)

fuj

O(n) iterations, but iteration takes more than O(1)

O(n) neighbors= O(n2)Could be fewer?

Each edge visited, times.2= O(E)

Size = O(E)Linear time.

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Graph Search

Which foundNotHandled node do we handle?

•Queue: Handle in order found.

•Breadth-First Search

•Stack: Handle most recently found

•Depth-First Search

•Priority Queue: Handle node that seems to be closest to s.

•Shortest (Weighted) Paths:

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Breadth First Search

<preCond> & <postCond>

Algorithm

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

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cfme

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

Queue

cfmej

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

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mejh

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

Queue

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BFS

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FoundNot Handled

Queue

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BFS

What order are the nodes found?

i.e. Breadth First Search

Algorithm

So far, the nodes have been found in order of length from s.

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

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BFS FoundNot Handled

Queue

d=0a

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d=1

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d=0d=1

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

Queue

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d=2

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BFS

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FoundNot Handled

Queued=0

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d=2

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BFS

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FoundNot Handled

Queued=0

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BFS

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FoundNot Handled

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d=2

d=2

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BFS

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FoundNot Handled

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d=2

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BFS

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FoundNot Handled

Queue

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d=2

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d=2

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BFS

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FoundNot Handled

Queue

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d=2

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BFS

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FoundNot Handled

Queue

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d=2

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BFS

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FoundNot Handled

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d=2

d=3

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BFS

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FoundNot Handled

Queue

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d=2

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d=3

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BFS

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FoundNot Handled

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d=4

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BFS

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FoundNot Handled

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d=3d=4

d=3

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

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BFS

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FoundNot Handled

Queued=0

d=1

d=2

d=3d=4

d=4d=5

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BFS

What order are the nodes found?

So far, the nodes have been found in order of length from s.

<postCond>: Finds a shortest path from sto each node v and its length.

To prove path is shortest:Prove there is a path of this length.Prove there are no shorter paths.

Give a path(witness)

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Basic Steps:

su

The shortest path to uhas length d

v

& there is an edge from u to v

There is a path to v with length d+1.

BFS

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BFS

What order are the nodes found?

So far, the nodes have been found in order of length from s.

<postCond>: Finds a shortest path from sto each node v and its length.

When we find v, we know there isn't a shorter path to it because ?

Prove there are no shorter paths.

Otherwise, we would have found it already.

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BFSData structure for storing tree: For each node v, store (v) to be parent of v.

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Basic Steps:

su

Path to u

v

& there is an edge from u to v

Parent of v is

(v)

(v) = u.

BFS

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Graph Search

Handle some foundNotHandled node

•Queue: Handle in order found.

•Breadth-First Search

•Stack: Handle most recently found

•Depth-First Search

•Priority Queue: Handle node that seems to be closest to s.

•Shortest (Weighted) Paths:

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Dijkstra's Shortest-Weighted Paths

Specification: Dijkstra's Shortest-Weighted Paths Reachability-from-single-source s

•<preCond>: The input is a graph G (either directed or undirected) with positive edge weights and a source node s. •<postCond>: Finds a shortest weighted path from s to each node v and its length.

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Dijkstra's Shortest-Weighted Paths

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100

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1w

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Length of shortest path from s to u?

4

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So far, the nodes have been found in order of length from s.

BFS

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100

11

1

1w

r

Which node is found first?

Is the same true for Dijkstra's Alg?

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So far, the nodes have been found in order of length from s.

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v

100

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1

1w

r

It has the longest path from s.Which node is found first?

Is the same true for Dijkstra's Alg?

BFS

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So far, the nodes have been found in order of length from s.

s

u

v

100

11

1

1w

r

In what order do we handle the foundNotHandled nodes?

handled

Dijkstra's

Handle node that “seems” to be closest to s.

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To prove path is shortest:Prove there is a path of this length.Prove there are no shorter paths.

Give a path(witness)

Dijkstra'sSo far, the nodes have been handled in order of length from s.

<postCond>: Finds a shortest weighted path from s to each node v and its length.

Desired

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To prove path is shortest:Prove there is a path of this length.Prove there are no shorter paths.

Dijkstra's

When we handle v, we know there isn't a shorter path to it because ?

Otherwise, we would have handled it already.

<postCond>: Finds a shortest weighted path from s to each node v and its length.

So far, the nodes have been handled in order of length from s.

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Dijkstra's

Handle node that “seems” to be closest to s.

Need to keep approximate shortest distances.•Path that we have “seen so far” will be calledhandled paths.•Let d(v) the length of the shortest such path to v.

Basic Steps:

uvs

Which is further?

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Basic Steps:

u

The shortest of handled paths to u has length d(u)

Dijkstra's

& there is an edge from u to v

w<u,v>

vs

The shortest of handled paths to v has length d(v)

The shortest known path to v has length

min( d(v), d(u)+w<u,v> ).

Updating d(u).

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…

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Dijkstra's

uvs

Handled nodes Found nodes

Handled Edges?

Definition of handled paths and d(v)

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Dijkstra's

uvs

Handled nodes Found nodes

Handled Edges

Handled Paths?

Definition of handled paths and d(v)

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Dijkstra's

uvs

Handled nodes Found nodes

Definition of handled paths and d(v)

Handled Edges

Handled Paths?

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Dijkstra's

uvs

Handled nodes

d(v) is the length of the shortest handled paths to v.

For handled u,d(u) is the length of the

shortest paths to u.

NotFound

d(v’)=

Definition of handled paths and d(v)

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Exit

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Dijkstra's

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Dijkstra's

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30

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d=

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Dijkstra's

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30

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2

3

3

85

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Dijkstra's

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d= d=

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30

152

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Dijkstra's

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d= d=

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3

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Dijkstra's

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d= d=

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d=5

30

1152

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3

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Dijkstra's

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d= d=

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30

1152

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3

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Dijkstra's

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d= d=

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30

1152

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3

90

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Dijkstra's

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d=3

d= d=

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d=4

d=5

30

1152

3

3

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Dijkstra's

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d= d=

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1152

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Dijkstra's

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d=4

d=5

30

1152

3

3

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Dijkstra's

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d=3

d= d=

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d=4

d=5

30

1152

3

DONE3

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Dijkstra'sFor handled w,

d(w) is the length of the shortest paths to w.

Handle node with smallest d(u).

For handled u, d(u) is the length of the shortest paths to u.

d(v) is the length of the shortest handled

path to v.

95

Dijkstra'sFor handled w,

d(w) is the length of the shortest paths to w.

Handle node with smallest d(u).

For handled u, d(u) is the length of the shortest paths to u.

d(u) is the length of the shortest handled

path to u.

d(u)

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Dijkstra'sFor handled w,

d(w) is the length of the shortest paths to w.

Handle node with smallest d(u).

For handled u, d(u) is the length of the shortest paths to u.

d(u)

d(u) is the length of the shortest handled

path to u.

First not handled node in path.

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Dijkstra'sFor handled w,

d(w) is the length of the shortest paths to w.

Handle node with smallest d(u).

For handled u, d(u) is the length of the shortest paths to u.

d(u)

First not handled node in path.d(u’)

d(u) & d(u’) are the length of the shortest handled

paths to u and u’.

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Dijkstra'sd(v) is the length

of the shortest handled path to v.

For handled w,d(w) is the length of the

shortest paths to w.

Handle node with smallest d(u).

d(v) is the length of the shortest handled path to v.

d(u)

d(v)

w<u,v>

d(v) = min( d(v), d(u)+w<u,v> ).

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Time = O(E)Each edge visited, times.2

?

Dijkstra's

This number of times following an edge.

Must update Priority Queue.

Takes time O(log(n))

Time = O(E log(n))

For each update, d(v) = min( d(v), d(u)+w<u,v> ).

Heap

100

Dijkstra's Shortest-Weighted PathsSpecification: Dijkstra's Shortest-Weighted Paths Reachability-from-single-source s

•<preCond>: The input is a graph G (either directed or undirected) with positive edge weights and a source node s. •<postCond>: Finds a shortest weighted path from s to each node v and its length.

EndingInitial ConditionsMake Progress

Maintain Loop InvDefine Exit ConditionDefine Step

Define Measure of Progress

Define Loop Invariants

Define Problem

km

79 km

to school

Exit

Exit

79 km 75 km

Exit

Exit

0 km Exit

101

Dijkstra's

102

Dijkstra's

103

Graph Search

Handle some foundNotHandled node

•Queue: Handle in order found.

•Breadth-First Search

•Stack: Handle most recently found

•Depth-First Search

•Priority Queue: Handle node that seems to be closest to s.

•Shortest (Weighted) Paths:

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DFS

Breadth first search makes a lot of sense for dating in general actually. It suggests dating a bunch of people casually before getting

serious rather than having a series of

five year relationships.

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DFS

s

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c

h

k

f

i

l

m

j

e

b

gd

FoundNot Handled

Stack

106

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

FoundNot Handled

Stack

s

107

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

FoundNot Handled

Stack

a

bgd

108

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

FoundNot Handled

Stack

a

mgd

e

109

DFS

110

DFS

The nodes in the stack form a path starting at s.

111

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

FoundNot Handled

Stack <node,# edges>

112

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

FoundNot Handled

Stack <node,# edges>

s,0

113

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,0

114

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,0

115

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,1h,0

116

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,1h,1k,0

117

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,1h,1

Tree Edge

Path on Stack

118

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,1

119

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,0

120

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,1

Cross Edgeto handled node

121

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,2

122

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,3l,0

123

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,3l,1

124

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,3

125

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4g,0

126

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4g,1j,0

127

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4g,1j,1

Back Edgeto node on Stack

Forms a cycle

128

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4g,1j,2m,0

129

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4g,1j,2m,1

130

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4g,1j,2

131

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4g,1

132

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,4

133

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,5f,0

134

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,5f,1

135

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2i,5

136

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,2

137

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1c,3

138

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,1

139

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

a,2

140

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,1

FoundNot Handled

Stack <node,# edges>

141

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,2

FoundNot Handled

Stack <node,# edges>

d,1

142

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,2

FoundNot Handled

Stack <node,# edges>

d,2

143

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,2

FoundNot Handled

Stack <node,# edges>

d,3e,0

144

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,2

FoundNot Handled

Stack <node,# edges>

d,3e,1

145

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,2

FoundNot Handled

Stack <node,# edges>

d,3

146

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,2

FoundNot Handled

Stack <node,# edges>

147

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,3

FoundNot Handled

Stack <node,# edges>

148

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,4

FoundNot Handled

Stack <node,# edges>

b,0

149

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,4

FoundNot Handled

Stack <node,# edges>

b,1

150

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,4

FoundNot Handled

Stack <node,# edges>

b,2

151

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,4

FoundNot Handled

Stack <node,# edges>

b,3

152

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

s,4

FoundNot Handled

Stack <node,# edges>

153

DFS

s

a

c

h

k

f

i

l

m

j

e

b

gd

FoundNot Handled

Stack <node,# edges>

done

154

155

RecursiveDepth First

Search

Get help from friends

156

157

Linear Order

underwear

pants

socks

shoes

underwearpantssocksshoes

socks underwearpantsshoes

158

Linear Order

underwear

pants

socks

shoes

?

159

Linear Order

a

b h

c i

d j

e k

f l

g

<preCond>: A Directed Acyclic Graph (DAG)

<postCond>: Find one valid linear order

….. l

Algorithm: •Find a sink.•Put it last in order.•Delete & Repeat

?

160

Linear Order

a

b h

c i

d j

e k

f l

g

<preCond>: A Directed Acyclic Graph (DAG)

<postCond>: Find one valid linear order

(n)(n2)

Algorithm: •Find a sink.•Put it last in order.•Delete & Repeat

….. l

161

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

degf

l

….. f

162

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

degl

l

When take off stackadd to backwards order

….. f

163

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

deg

l

When take off stackadd to backwards order

l,f

164

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

del

When take off stackadd to backwards order

g,l,f

165

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

dl

When take off stackadd to backwards order

e,g,l,f

166

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

l

When take off stackadd to backwards order

d,e,g,l,f

167

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

il

When take off stackadd to backwards order

d,e,g,l,f

jk

168

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

il

When take off stackadd to backwards order

k,d,e,g,l,f

j

169

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

il

When take off stackadd to backwards order

j,k,d,e,g,l,f

170

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

l

When take off stackadd to backwards order

i,j,k,d,e,g,l,f

171

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

bl

When take off stackadd to backwards order

c

i,j,k,d,e,g,l,f

172

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

bl

When take off stackadd to backwards order

c,i,j,k,d,e,g,l,f

173

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

l

When take off stackadd to backwards order

b,c,i,j,k,d,e,g,l,f

174

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

al

When take off stackadd to backwards order

h

b,c,i,j,k,d,e,g,l,f

175

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

al

When take off stackadd to backwards order

h,b,c,i,j,k,d,e,g,l,f

176

Linear Order

a

b h

c i

d j

e k

f

g

FoundNot Handled

Stack

Alg: DFS

l

When take off stackadd to backwards order

a,h,b,c,i,j,k,d,e,g,l,f done

177

Linear OrderFound

Not HandledStack

Proof:•Case 1: u goes on stack first before v.

•Because of edge, v goes on before u comes off•v comes off before u comes off•v goes after u in order.

u vv…u…

Consider each edge

v

…

u

…

178

Linear OrderFound

Not HandledStack

Proof:•Case 1: u goes on stack first before v.•Case 2: v goes on stack first before u. v comes off before u goes on.

•v goes after u in order.

u vv…u…

Consider each edge

u

…

v

…

179

Linear OrderFound

Not HandledStack

Proof:•Case 1: u goes on stack first before v.•Case 2: v goes on stack first before u. v comes off before u goes on.Case 3: v goes on stack first before u. u goes on before v comes off.

•Panic: u goes after v in order. •Cycle means linear order is impossible

u vu…v…

Consider each edge

u

…

v

…

The nodes in the stack form a path starting at s.

done

180

End