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UNIVERSITY OF SOUTH CAROLINA Department of Computer Science andEn ineerin

CSCE 580Artificial IntelligenceDijkstras Algorithm: Notes to

Complement and Reinforcethe Graduate Student

PresentationFall 2008

Marco Valtorta

[email protected]

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Example: Romania

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Single-State ProblemFormulation

A problem is defined by four items:

1. initial state e.g., "at Arad"2. actions or successor functionS(x) = set of action

state pairs e.g., S(Arad) = {,

, }3. goal test, can be

explicit, e.g.,x= "at Bucharest" implicit, e.g., Checkmate(x)

4. path cost (additive) e.g., sum of distances, number of actions

executed, etc. c(x,a,y) is the step cost, assumed to be 0

A solution is a sequence of actions leading fromthe initial state to a goal state

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Uniform-Cost (Dijkstra) forGraphs

Original Reference: Dijkstra, E. W. "A Note on Two Problems inConnexion with Graphs. Numerische Matematik, 1 (1959),269-271

1. Put the start node s in OPEN. Set g(s) to 0 2. If OPEN is empty, exit with failure

3. Remove from OPEN and place in CLOSED a node n forwhich g(n) is minimum; in case of ties, favor a goal node

4. If n is a goal node, exit with the solution obtained bytracing back pointers from n to s

5. Expand n, generating all of its successors. For eachsuccessor n' of n:

a. compute g'(n')=g(n)+c(n,n') b. if n' is already on OPEN, and g'(n')

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Properties of DijkstrasAlgorithm

g(n) is the distance (cost) of some path from the startnode to node n

f*(n) is the minimum of g(n) over all possible paths fromthe start node to node n

f*(k) is the distance of the goal node from the start

node---assume a single goal node, k, for simplicity A search algorithm is admissible if it returns the

shortest path

1. When node n is closed in step 3, g(n)=f*(n)

2. When node n is closed at step 3, g(n) is at least as

high as the g value of all already closed nodes and allnodes with lower g values than g(n) have already beenclosed

3. Only nodes for which g(n)

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Property 1 of DijkstrasAlgorithm

When node n is closed in step 3, g(n)=f*(n) The proof is by induction on the number of

closed nodes. For the base case, g(s) = f*(s) =0.

Let n1,,nq be the nodes in OPEN. Let n = ni Consider the path to n through nodes in

CLOSED that establishes that n is to be closedin step 3

Assume that there is a shorter path to n thanthe one chosen by Dijkstras algorithm. Let npbe the first node in OPEN on that path. Sincepath snpni is shorter than g(n), then pathsnp is shorter than path sni.This means thatwhen executing step 3, g(np) < g(ni), andtherefore np would be closed in step 3 instead

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7/11UNIVERSITY OF SOUTH CAROLINA Department of Computer Science andEn ineerin


When node n is closed at step 3, g(n) is atleast as high as the g value of all alreadyclosed nodes and all nodes with lower g

values than g(n) have already beenclosed

This can also be proven by induction. For theinduction step, note that the node that is

closed at step 3 has g(n) value that is at leastas high than that of all previously closednodes---otherwise it would have been closedbefore!

Also note that this proof depends crucially onhaving non-negative edge costs

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Property 3 and 4 of DijkstrasAlgorithm

Only nodes for which g(n)

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Dijkstra's algorithm expands the least numberof nodes among all admissible blind,

unidirectional search algorithms

Adversary argument

I claim there is an algorithm (say, B) that does notexpand node m, for which g(m) < f*(k) on somegraph search problem (say, G) and is admissible

Let f*(k)-g(m) = e

Consider an instance of graph search problemidentical to G except for the additional edge mkof cost e/2

Since B does not expand node m, it will not returnthe shortest path of length f*(k)-(e/2)

Therefore, B is not admissible

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Lower Bound Assumptions and notation:

decision-tree model (count comparisons of edge costs) blind unidirectional algorithms

n is the number of nodes and m is the number of edgesin the (implicit) graph been searched

Property 5 implies (m) (n log(n)) follows from a transformation of

sorting to finding the shortest path spanning treeand a sequence of all the n nodes ordered inincreasing distance from the start node

Overall: (m + n log(n))

The Fibonacci tree implementation of Dijkstrasalgorithm matches the lower bound and istherefore optimal

See reference in notes

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Bidirectional Uniform-CostAlgorithm (Assume that there is only one goal node, k.)

1. Put the start node s in OPEN1 and the goal node k in OPEN2. Set g(s) andh(k) to 0

2'. If OPEN1 is empty, exit with failure 3'. Remove from OPEN1 and place in CLOSED1 a node n for which g(n) is

minimum 4'. If n is in CLOSED2, exit with the solution obtained by tracing backpointers

from n to s and forward pointers from n to k 5'. Expand n, generating all of its successors. For each successor n' of n:

a. compute g'(n')=g(n)+c(n,n') b. if n' is already on OPEN1, and g'(n')

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