Shortest Paths

08-07-2006PowerPoint adapted from Alan Tam’s Shortest Path, 2004with slight modifications

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Breadth First Search (BFS)

Techniques to Enumerate all Vertices for Goal

BFS needs O(V) queue and O(V) set for duplicate elimination and runs in O(V + E) time

BFS can find Shortest Path if Graph is Not Weighted

BFS works because a Queue ensures a specific Order

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What Order?

Define d[v] be the length of shortest path from s to v

At any time, vertices are classified into: Black With known d[v] Grey With some known path White Not yet touched

The Grey vertex with shortest known path has potential to become Black

The Head of Queue is always the case in an unweighted graph

Why?

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Weighted Graph

Queue does not promise smallest d[v] anymore

Expanding weighted edge causeds unnecessary searching of artificial vertices

We can simply pick the real vertex nearest to the starting vertex

We need “Sorted Queue” which “dequeues” vertices in increasing order of d[v].

It is called a “Priority Queue” and negation of d[v] is called the Priority of vertex v

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Dijkstra’s Algorithm

d ← ∞d[s] ← 0enqueue(queue, s)while not empty(queue) do

u = dequeue(queue)for-each v where (u, v) in E

if d[v] > d[u] + wuv thend[v] = d[u] + wuv

enqueue(queue, v)

Note that d[v] represents the known shortest distance of v from s during the process and may not be the real shortest distance until v is marked black.

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Requirements for Priority Queue Map<V, int>

Store temporary values of d[v] PriorityQueue<V, function: V→int >

Extract vertices in increasing order of d[v] Initially all d[v] are infinity When we visit a White vertex

Set d[v] Insert v into Queue

When we visit a Gray vertex Update d[v] What to do to the Priority Queue?

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Implementing Priority Queue? Usage:

Insert: V Extract: V Decrease: E - V

Array Insert: O(1) Extract: O(n) Decrease: O(1)

Sorted Array Insert: O(n) Extract: O(1) Decrease: O(n)

Heap Insert: O(lg n) Extract: O(lg n) Decrease: O(lg n)

Fibonacci Heap Insert: O(1) Extract: Amortized O(lg

n) Decrease: O(1)

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Time Complexity

Worst Case Big-Oh Memory Time

Array V

Array (Lazy)

Sorted Array

Heap

Heap (Lazy)

Fibonacci Heap V E + V lg V

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Time Complexity

Worst Case Big-Oh Memory Time

Array V V2

Array (Lazy) E VE

Sorted Array V VE

Heap V E lg V

Heap (Lazy) E E lg V

Fibonacci Heap V E + V lg V

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Another Attempt

Consider the following code segment:For each edge (u, v)

If d[v] > d[u] + wuv

d[v] ← d[u] + wuv Assume one of the shortest paths is

(s, v1, v2, …, vk) If d[vi] = its shortest path from s After this loop, d[vi+1] = its shortest path from s By MI, After k such loops, found shortest path from s

to vk

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Bellman-Ford Algorithm

All v1, v2, …,vk distinct?

Do V-1 timesfor-each (u, v) in Eif d[v] > d[u] + wuv then

d[v] ← d[u] + wuv

Order = O(VE) Support Negative-weight Edges

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Another Attempt

Number vertices as v1, v2, …, vV Define no≤k-path as a path which does not pass

through vertices v1, v2, …, vk A path s-t must either be

a no≤1-path, or concatenation of a no≤1-path s-v1 and a no≤1-path v1-t

A no≤1-path s-t must either be a no≤2-path, or concatenation of a no≤2-path s-v2 and a no≤2-path v2-t

By MI …

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Trivial Paths

S TFor all

SFor all

TDirect Pathfrom S to T

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R SFor all

RFor all

SDirect Pathfrom R to SS T

For all S

For all TDirect Path

from S to T

For S = vV

Direct Pathfrom R to T

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no≤(V-1)-Paths

S TFor all

SFor all

TShortest Pathfrom S to T

via vV

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R SFor all

RFor all

SShortest Pathfrom R to S

via vV

S TFor all

SFor all

TShortest Pathfrom S to T

via vV

For S = vV-1

Shortest Pathfrom R to T

via vV

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no≤(V-2)-Paths

S TFor all

SFor all

TShortest Pathfrom S to Tvia vV-1,vV

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no≤2-Paths

S TFor all

SFor all

TShortest Pathfrom S to T

via v3,v4,...,vV

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R SFor all

RFor all

SShortest Pathfrom R to S

via v3,v4,...,vV

S TFor all

SFor all

TShortest Pathfrom S to Tvia v3,v4,...,vV

For S = V2

Shortest Pathfrom R to Tvia v3,v4,...vV

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no≤1-Paths

S TFor all

SFor all

TShortest Pathfrom S to T

via v2,v3,...,vV

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R SFor all

RFor all

SShortest Pathfrom R to S

via v2,v3,...,vV

S TFor all

SFor all

TShortest Pathfrom S to Tvia v2,v3,...,vV

For S = V1

Shortest Pathfrom R to Tvia v2,v3,...vV

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Shortest Paths

S TFor all

SFor all

TShortest Pathfrom S to T

via v1,v2,...,vV

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Warshall-Floyd Algorithm

for-each (u, v) in Ed[u][v] ← wuv

for-each i in Vfor-each j in V

for-each k in Vif d[j][k] > d[j][i] + d[i][k]

d[j][k] ← d[j][i] + d[i][k]

Time Complexity: O(V3)

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Graph Modeling

Conversion of a problem into a graph problem

Sometimes a problem can be easily solved once its underlying graph model is recognized

Graph modeling appears almost every year in NOI or IOI

(cx, 2004)

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Basics of Graph Modeling

A few steps: identify the vertices and the edges identify the objective of the problem state the objective in graph terms implementation:

• construct the graph from the input instance• run the suitable graph algorithms on the graph• convert the output to the required format

(cx, 2004)