Algorithms and Data Structures. . . . . .1/1 Algorithms and Data Structures 12th Lecture: Graph...

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Algorithms and Data Structures12th Lecture: Graph Algorithms II

Yutaka Watanobe, Jie Huang, Yan Pei,Wenxi Chen, Qiangfu Zhao, Wanming Chu

University of Aizu

Last Updated: 2020/12/1

Y. Watanobe, J. Huang, Y. Pei, W. Chen, Q. Zhao, W. Chu University of Aizu

Algorithms and Data Structures

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Outline

Weighted GraphsMinimum Spanning TreeSingle-Source Shortest Paths



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Adjacency Matrix Representation: UndirectedWeighted Graph

For a given connected, undirected graph G = (V ,E) where foreach edge (u, v) ∈ E , we have a weight w(u, v) specifying thecost to connect u and v .

0 1 2 3 4 5 6

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7 -2

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∞ ∞ ∞

∞ ∞ ∞

∞ ∞ ∞

∞ ∞ ∞

∞ ∞ ∞

∞ ∞ ∞ ∞

∞ ∞∞ ∞ ∞

∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞

∞ ∞

∞



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Adjacency List Representation: Directed WeightedGraph

For a given connected, directed graph G = (V ,E) where for eachedge (u, v) ∈ E , we have a weight w(u, v) specifying the cost toconnect u and v .

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Spanning Tree

(a) (b) (c)



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Minimum Spanning Tree (1)

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We wish to find an acyclic subset T ⊆ E that connects all of thevertices and whose total weight

w(T ) =∑

(u,v)∈T

w(u, v)

is minimized.Since T is acyclic and connects all of the vertices, it must form atree, which we call a spanning tree.We call the problem of determining the tree T the minimumspanning-tree (MST) problem.



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A minimum spanning tree for a connected graph.The weights on edges are shown, and the edges in a minimumspanning tree are shaded.The total weight of the tree is 37.



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Growing a Minimum Spanning Tree (1)

Assume that we have a connected, undirected graph G = (V ,E)with a weight function w : E → R, and we wish to find a MST forG.We use a greedy approach. It is captured by the ”generic”algorithm, which grows the minimum spanning tree one edge ata time.The algorithm manages a set of edges T , maintaining thefollowing loop invariant:

Prior to each iteration, T is a subset of some minimum spanningtree.



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Growing a Minimum Spanning Tree (2)

At each step, we determine an edge (u, v) that can be added toT without violating this invariant, in the sense that T ∪ {(u, v)} isalso a subset of a MST.We call such an edge a safe edge for T , since it can be safelyadded to T while maintaining the invariant.



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Generic Algorithm for MST

genericMST(G, w)

T = empty

while T does not form a spanning tree

find an edge (u, v) that is safe for T

T = T ∪ {(u, v)}

return T

This is a “generic” algorithm which can be a basis to implementdifferent algorithms.



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Prim’s Algorithm for MST

Prim’s algorithm is a special case of the generic MST algorithms.Prim’s algorithm has the property that the edges in the set Talways form a single tree.The tree starts from an arbitrary root vertex r and grows until thetree spans all the vertices in V .At each step, a light edge is added to the tree T that connects Tto an isolated vertex of GT = (V ,T ). This rule adds only edgesthat are safe for T .This strategy is greedy since the tree is augmented at each stepwith an edge that contributes the minimum amount possible tothe tree’s weight.



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Prim’s Algorithm based on the Generic Algorithm (1)

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Prim’s Algorithm: Implementation

T T’

V - T V - T

u

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Candidates

r r

Update



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Prim’s Algorithm1

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Prim’s Algorithm (1)

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Prim’s Algorithm: O(|V |2) Implementation (1)

prim(G, w, r)

for each u in G

d[u] = inf

pi[u] = NIL

color[u] = WHITE

d[r] = 0



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Prim’s Algorithm: O(|V |2) Implementation (2)

while true

mincost = inf

for each i in G

if color[i] != BLACK and d[i] < mincost

mincost = d[i]

u = i

if mincost == inf

break

color[u] = BLACK

for each v in Adj[u]

if color[v] != BLACK and w(u, v) < d[v]

pi[v] = u

d[v] = w(u, v)



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Prim’s algorithm: ((|V |+ |E |) log |V |) Implementation(1)

prim(G, w, r)

for each u in G

d[u] = inf

pi[u] = NIL



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Prim’s algorithm: ((|V |+ |E |) log |V |) Implementation(2)

For an efficient implementation, we use a min-heap H of vertices,keyed by their d values.

d[r] = 0

H = buildMinHeap(V)

while H is not empty

u = H.extractMin()

color[u] = BLACK


if color[v] != BLACK

if w(u, v) < d[v]

pi[v] = u

d[v] = w(u, v)

heapDecreaseKey(H, v, w(u, v))



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Single-Source Shortest Paths (1)

We are given a weighted, directed graph G = (V ,E), with weightfunction w : E → R mapping edges to real-valued weights.The weight of path p = {v0, v1, ..., vk} is the sum of the weights ofits constituent edges:

w(p) =k∑

i=1

w(vi−1, vi)



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Single-Source shortest Paths (2)

We define the shortest-path weight from u to v by

δ(u, v) ={

min{w(p) : u → v} if there is a path from u to v∞ otherwise

A shortest path from vertex u to vertex v is then defined as anypath p with weight w(p) = δ(u, v).Given a graph G = (V ,E), we want to find a shortest path from agiven source vertex s ∈ V to each vertex v ∈ V .



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Initialize

For each vertex v ∈ V , we maintain an attribute d [v ], which is anupper bound on the weight of a shortest path from source s to v .We call d [v ] a shortest-path estimate.We initialize the shortest-path estimates and predecessors bythe following procedure.

initializeSingleSource(G, s)

for each v in G

d[v] = inf

pi[v] = NIL

d[s] = 0



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Relaxation

The process of relaxing an edge (u, v) consists of testingwhether we can improve the shortest path to v found so far bygoing through u and, if so, updating d [v ] and pi[v ].A relaxation step may decrease the value of the shortest-pathestimate d [v ] and update v ’s predecessor field pi[v ].

relax(u, v, w)

if d[v] > d[u] + w(u, v)

d[v] = d[u] + w(u, v)

pi[v] = u

u

s

v

w(u, v)

d[v]

d[u]



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Dijkstra’s Algorithm for SSSP (1)

Dijkstra’s algorithm solves the single-source shortest-pathsproblem on a weighted, directed graph G = (V ,E) for the case inwhich all edge weights are nonnegative.Dijkstra’s algorithm maintains a set S of vertices whose finalshortest path weights from the source s have already beendetermined.



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Dijkstra’s Algorithm for SSSP (2)

The algorithm repeatedly selects the vertex u ∈ V − S with theminimum shortest-path estimate, adds u to S, and relaxes alledges leaving u.

S S’

V - S V - S

u

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Candidates

s s

Update



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Dijkstra’s Algorithm: O(|V |2) Implementation

dijkstra(G, w, s)


while true

mincost = inf

for each i in G

if color[i] != BLACK and d[i] < mincost

mincost = d[i]

u = i

if mincost == inf

break

color[u] = BLACK


if color[v] != BLACK and d[u] + w(u, v) < d[v]

pi[v] = u

d[v] = d[u] + w(u, v)



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Dijkstra’s Algorithm: O((|V |+ |E |) log |V |)Implementation

For an efficient implementation, we use a min-heap H of vertices,keyed by their d values.

dijkstra(G, w, s)


H = buildMinHeap(V)

while H is not empty

u = H.extractMin()

color[u] = BLACK


if color[v] != BLACK

if d[u] + w(u, v) < d[v]

d[v] = d[u] + w(u, v)

color[v] = GRAY

hepaDecreaseKey(H, v, d[u] + w(u, v))



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Dijkstra’s Algorithm (1)

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Dijkstra’s Algorithm: Another Example1

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Dijkstra’s Algorithm: Another Example (1)

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Reference

1 Introduction to Algorithms (third edition), Thomas H.Cormen,Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. TheMIT Press, 2012.



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