CS 477/677 - Lecture 20 1

Minimum Spanning TreesProblem• A town has a set of houses

and a set of roads• A road connects 2 and only

2 houses• A road connecting houses u and v has a repair cost

w(u, v)Goal: Repair enough (and no more) roads such

that:1. Everyone stays connected: can reach every house

from all other houses, and2. Total repair cost is minimum

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CS 477/677 - Lecture 20 2

Minimum Spanning Trees

• A connected, undirected graph:

– Vertices = houses, Edges = roads

• A weight w(u, v) on each edge (u, v) E

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Find T E such that:

1. T connects all vertices

2. w(T) = Σ(u,v)T w(u, v) is

minimized

CS 477/677 - Lecture 20 3

Minimum Spanning Trees

• T forms a tree = spanning tree • A spanning tree whose weight is minimum over

all spanning trees is called a minimum spanning tree, or MST.

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CS 477/677 - Lecture 20 4

Properties of Minimum Spanning Trees

• Minimum spanning trees are not unique– Can replace (b, c) with (a, h) to obtain a different

spanning tree with the same cost

• MST have no cycles– We can take out an edge of

a cycle, and still have the

vertices connected while reducing the cost

• # of edges in a MST:– |V| - 1

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CS 477/677 - Lecture 20 5

Growing a MST

Minimum-spanning-tree problem: find a MST for a connected, undirected graph, with a weight function associated with its edges

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A generic solution:• Build a set A of edges (initially

empty)

• Incrementally add edges to A such that they would belong to a MST

• An edge (u, v) is safe for A if and only if A {(u, v)} is also a subset of some MST

We will add only safe edges

CS 477/677 - Lecture 20 6

Generic MST algorithm

1. A ←

2. while A is not a spanning tree

3. do find an edge (u, v) that is safe for A

4. A ← A {(u, v)}

5. return A

• How do we find safe edges?

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CS 477/677 - Lecture 20 7

We would addedge (c, f)

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The Algorithm of Kruskal

• Start with each vertex being its own component

• Repeatedly merge two components into one by choosing the light edge that connects them

• Scan the set of edges in monotonically increasing order by weight

• Uses a disjoint-set data structure to determine whether an edge connects vertices in different components

CS 477/677 - Lecture 20 8

Operations on Disjoint Data Sets

• MAKE-SET(u) – creates a new set whose only member is u

• FIND-SET(u) – returns a representative element from the set that contains u– May be any of the elements of the set that has a

particular property

– E.g.: Su = {r, s, t, u}, the property is that the element

be the first one alphabetically

FIND-SET(u) = r FIND-SET(s) = r– FIND-SET has to return the same value for a given set

CS 477/677 - Lecture 20 9

Operations on Disjoint Data Sets

• UNION(u, v) – unites the dynamic sets that

contain u and v, say Su and Sv

– E.g.: Su = {r, s, t, u}, Sv = {v, x, y}

UNION (u, v) = {r, s, t, u, v, x, y}

CS 477/677 - Lecture 20 10

KRUSKAL(V, E, w)1. A ← 2. for each vertex v V3. do MAKE-SET(v)4. sort E into non-decreasing order by weight w5. for each (u, v) taken from the sorted list6. do if FIND-SET(u) FIND-SET(v)7. then A ← A {(u, v)} 8. UNION(u, v)9. return ARunning time: O(E lgV) – dependent on the implementation

of the disjoint-set data structure

CS 477/677 - Lecture 20 11

Example

1. Add (h, g)2. Add (c, i)3. Add (g, f)4. Add (a, b)5. Add (c, f)6. Ignore (i, g)7. Add (c, d)8. Ignore (i, h)9. Add (a, h)10. Ignore (b, c)11. Add (d, e)12. Ignore (e, f)13. Ignore (b, h)14. Ignore (d, f)

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1: (h, g)2: (c, i), (g, f)4: (a, b), (c, f)6: (i, g)7: (c, d), (i, h)

8: (a, h), (b, c) 9: (d, e)10: (e, f)11: (b, h)14: (d, f)

{g, h}, {a}, {b}, {c}, {d}, {e}, {f}, {i}

{g, h}, {c, i}, {a}, {b}, {d}, {e}, {f}

{g, h, f}, {c, i}, {a}, {b}, {d}, {e}

{g, h, f}, {c, i}, {a, b}, {d}, {e}

{g, h, f, c, i}, {a, b}, {d}, {e}

{g, h, f, c, i}, {a, b}, {d}, {e}

{g, h, f, c, i, d}, {a, b}, {e}

{g, h, f, c, i, d}, {a, b}, {e}

{g, h, f, c, i, d, a, b}, {e}

{g, h, f, c, i, d, a, b}, {e}

{g, h, f, c, i, d, a, b, e}

{g, h, f, c, i, d, a, b, e}

{g, h, f, c, i, d, a, b, e}

{g, h, f, c, i, d, a, b, e}{a}, {b}, {c}, {d}, {e}, {f}, {g}, {h}, {i}

CS 477/677 - Lecture 20 12

The algorithm of Prim

• The edges in set A always form a single tree

• Starts from an arbitrary “root”: VA = {a}

• At each step:

– Find a light edge crossing cut (VA, V - VA)

– Add this edge to A

– Repeat until the tree spans all vertices

• Greedy strategy– At each step the edge added contributes the minimum amount

possible to the weight of the tree

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CS 477/677 - Lecture 20 13

How to Find Light Edges Quickly?Use a priority queue Q:

• Contains all vertices not yet

included in the tree (V – VA)– V = {a}, Q = {b, c, d, e, f, g, h, i}

• With each vertex we associate a key:

key[v] = minimum weight of any edge (u, v) connecting v to a vertex in the tree

– Key of v is if v is not adjacent to any vertices in VA

– After adding a new node to VA we update the weights of all the nodes

adjacent to it

– We added node a key[b] = 4, key[h] = 8

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CS 477/677 - Lecture 20 14

PRIM(V, E, w, r)1. Q ← 2. for each u V

3. do key[u] ← ∞

4. π[u] ← NIL

5. INSERT(Q, u)

6. DECREASE-KEY(Q, r, 0)

7. while Q

8. do u ← EXTRACT-MIN(Q)

9. for each v Adj[u]10. do if v Q and w(u, v) < key[v]11. then π[v] ← u12. DECREASE-KEY(Q, v, w(u, v))

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0

0 Q = {a, b, c, d, e, f, g, h, i} VA =

Extract-MIN(Q) a

CS 477/677 - Lecture 20 15

Example 0

Q = {a, b, c, d, e, f, g, h, i} VA =

Extract-MIN(Q) a

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key [b] = 4 [b] = akey [h] = 8 [h] = a

4 8 Q = {b, c, d, e, f, g, h, i} VA = {a}

Extract-MIN(Q) b

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8

CS 477/677 - Lecture 20 16

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Example

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key [c] = 8 [c] = bkey [h] = 8 [h] = a - unchanged

8 8 Q = {c, d, e, f, g, h, i} VA = {a, b}

Extract-MIN(Q) c

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key [d] = 7 [d] = ckey [f] = 4 [f] = ckey [i] = 2 [i] = c

7 4 8 2 Q = {d, e, f, g, h, i} VA = {a, b, c}

Extract-MIN(Q) i

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CS 477/677 - Lecture 20 17

Example

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key [h] = 7 [h] = ikey [g] = 6 [g] = i

7 4 8 Q = {d, e, f, g, h} VA = {a, b, c, i}

Extract-MIN(Q) f

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key [g] = 2 [g] = fkey [d] = 7 [d] = c unchanged

key [e] = 10 [e] = f 7 10 2 8

Q = {d, e, g, h} VA = {a, b, c, i, f}

Extract-MIN(Q) g

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CS 477/677 - Lecture 20 18

Example

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key [h] = 1 [h] = g 7 10 1 Q = {d, e, h} VA = {a, b, c, i, f, g}

Extract-MIN(Q) h

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7 10 Q = {d, e} VA = {a, b, c, i, f, g, h}

Extract-MIN(Q) d

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CS 477/677 - Lecture 20 19

Example

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key [e] = 9 [e] = f 9 Q = {e} VA = {a, b, c, i, f, g, h, d}

Extract-MIN(Q) eQ = VA = {a, b, c, i, f, g, h, d, e}

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CS 477/677 - Lecture 20 20

PRIM(V, E, w, r)1. Q ← 2. for each u V

3. do key[u] ← ∞

4. π[u] ← NIL

5. INSERT(Q, u)

6. DECREASE-KEY(Q, r, 0) ► key[r] ← 07. while Q

8. do u ← EXTRACT-MIN(Q)

9. for each v Adj[u]10. do if v Q and w(u, v) < key[v]11. then π[v] ← u12. DECREASE-KEY(Q, v, w(u, v))

O(V) if Q is implemented as a min-heap

Executed |V| times

Takes O(lgV)

Min-heap operations:O(VlgV)

Executed O(E) times

Constant

Takes O(lgV)

O(ElgV)

Total time: O(VlgV + ElgV) = O(ElgV)

CS 477/677 - Lecture 20 21

Shortest Path Problems

• How can we find the shortest route between two points on a map?

• Model the problem as a graph problem:– Road map is a weighted graph:

vertices = cities

edges = road segments between cities

edge weights = road distances

– Goal: find a shortest path between two vertices (cities)

CS 477/677 - Lecture 20 22

Shortest Path Problems

• Input:– Directed graph G = (V, E)

– Weight function w : E → R

• Weight of path p = v0, v1, . . . , vk

• Shortest-path weight from u to v:

δ(u, v) = min w(p) : u v if there exists a path from u to v

∞ otherwise

• Shortest path u to v is any path p such that w(p) = δ(u, v)

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CS 477/677 - Lecture 20 23

Variants of Shortest Paths

• Single-source shortest path– G = (V, E) find a shortest path from a given source vertex s to

each vertex v V• Single-destination shortest path

– Find a shortest path to a given destination vertex t from each vertex v

– Reverse the direction of each edge single-source

• Single-pair shortest path– Find a shortest path from u to v for given vertices u and v– Solve the single-source problem

• All-pairs shortest-paths– Find a shortest path from u to v for every pair of vertices u and v

CS 477/677 - Lecture 20 24

Optimal Substructure of Shortest Paths

Given:– A weighted, directed graph G = (V, E)

– A weight function w: E R,

– A shortest path p = v1, v2, . . . , vk from v1 to vk

– A subpath of p: pij = vi, vi+1, . . . , vj, with 1 i j k

Then: pij is a shortest path from vi to vj

Proof: p = v1 vi vj vk

w(p) = w(p1i) + w(pij) + w(pjk) Assume pij’ from vi to vj with w(pij’) < w(pij) w(p’) = w(p1i) + w(pij’) + w(pjk) < w(p) contradiction!

p1i pij pjk

v1

vi

vj

vk

p1i

pij

pij’

pjk

CS 477/677 - Lecture 20 25

Negative-Weight Edges

• s a: only one path

(s, a) = w(s, a) = 3

• s b: only one path

(s, b) = w(s, a) + w(a, b) = -1

• s c: infinitely many paths

s, c, s, c, d, c, s, c, d, c, d, c

cycle has positive weight (6 - 3 = 3)

s, c is shortest path with weight (s, b) = w(s, c) = 5

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weight edges?

CS 477/677 - Lecture 20 26

Negative-Weight Edges• s e: infinitely many paths:

s, e, s, e, f, e, s, e, f, e, f, e– cycle e, f, e has negative

weight: 3 + (- 6) = -3

– can find paths from s to e with arbitrarily large negative weights

(s, e) = - no shortest path exists between s and e

– Similarly: (s, f) = - , (s, g) = -

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CS 477/677 - Lecture 20 27

Negative-Weight Edges

• Negative-weight edges may form

negative-weight cycles

• If such cycles are reachable from

the source: (s, v) is not properly

defined

– Keep going around the cycle, and get

w(s, v) = - for all v on the cycle

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CS 477/677 - Lecture 20 28

Cycles

• Can shortest paths contain cycles?

• Negative-weight cycles

• Positive-weight cycles:– By removing the cycle we can get a shorter path

• Zero-weight cycles– No reason to use them

– Can remove them to obtain a path with similar weight

• We will assume that when we are finding shortest paths, the paths will have no cycles

No!No!

CS 477/677 - Lecture 20 29

Shortest-Path Representation

For each vertex v V:• d[v] = δ(s, v): a shortest-path estimate

– Initially, d[v]=∞– Reduces as algorithms progress

[v] = predecessor of v on a shortest path from s– If no predecessor, [v] = NIL induces a tree—shortest-path tree

• Shortest paths & shortest path trees are not unique

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CS 477/677 - Lecture 20 30

Initialization

Alg.: INITIALIZE-SINGLE-SOURCE(V, s)

1. for each v V

2. do d[v] ←

3. [v] ← NIL

4. d[s] ← 0

• All the shortest-paths algorithms start with INITIALIZE-SINGLE-SOURCE

CS 477/677 - Lecture 20 31

Relaxation• Relaxing an edge (u, v) = testing whether we

can improve the shortest path to v found so far by going through u

If d[v] > d[u] + w(u, v) we can improve the shortest path to v update d[v] and [v]

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RELAX(u, v, w)

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RELAX(u, v, w)

After relaxation:d[v] d[u] + w(u, v)

s s

CS 477/677 - Lecture 20 32

RELAX(u, v, w)

1. if d[v] > d[u] + w(u, v)2. then d[v] ← d[u] + w(u, v)3. [v] ← u

• All the single-source shortest-paths algorithms – start by calling INIT-SINGLE-SOURCE– then relax edges

• The algorithms differ in the order and how many times they relax each edge

CS 477/677 - Lecture 20 33

Bellman-Ford Algorithm

• Single-source shortest paths problem– Computes d[v] and [v] for all v V

• Allows negative edge weights• Returns:

– TRUE if no negative-weight cycles are reachable from the source s

– FALSE otherwise no solution exists• Idea:

– Traverse all the edges |V – 1| times, every time performing a relaxation step of each edge

CS 477/677 - Lecture 20 34

BELLMAN-FORD(V, E, w, s)

1. INITIALIZE-SINGLE-SOURCE(V, s)2. for i ← 1 to |V| - 13. do for each edge (u, v) E4. do RELAX(u, v, w)5. for each edge (u, v) E6. do if d[v] > d[u] + w(u, v)7. then return FALSE8. return TRUE

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E: (t, x), (t, y), (t, z), (x, t), (y, x), (y, z), (z, x), (z, s), (s, t), (s, y)

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CS 477/677 - Lecture 20 35

Example

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Pass 1 Pass 2

Pass 3 Pass 4

CS 477/677 - Lecture 20 36

Detecting Negative Cycles

• for each edge (u, v) E• do if d[v] > d[u] + w(u, v)• then return FALSE• return TRUE

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d[b] = -1d[s] + w(s, b) = -4

d[b] > d[s] + w(s, b)

CS 477/677 - Lecture 20 37

BELLMAN-FORD(V, E, w, s)

1. INITIALIZE-SINGLE-SOURCE(V, s)2. for i ← 1 to |V| - 13. do for each edge (u, v) E4. do RELAX(u, v, w)5. for each edge (u, v) E6. do if d[v] > d[u] + w(u, v)7. then return FALSE8. return TRUE

Running time: O(VE)

(V)O(V)

O(E)

O(E)

CS 477/677 - Lecture 20 38

Shortest Path Properties

• Triangle inequalityFor all (u, v) E, we have:

δ(s, v) ≤ δ(s, u) + w(u, v)

• If u is on the shortest path to v we have the equality sign

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CS 477/677 - Lecture 20 39

Shortest Path Properties• Upper-bound property

We always have d[v] ≥ δ(s, v) for all v.

Once d[v] = δ(s, v), it never changes.– The estimate never goes up – relaxation only lowers the

estimate

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CS 477/677 - Lecture 20 40

Shortest Path Properties

• No-path propertyIf there is no path from s to v then d[v] = ∞ always.– δ(s, h) = ∞ and d[h] ≥ δ(s, h) d[h] = ∞

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CS 477/677 - Lecture 20 41

Shortest Path Properties

• Convergence propertyIf s u → v is a shortest path, and if d[u] = δ(s, u) at any time prior to relaxing edge (u, v), then d[v] = δ(s, v) at all times afterward.

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• If d[v] > δ(s, v) after relaxation:d[v] = d[u] + w(u, v)

d[v] = 5 + 2 = 7• Otherwise, the value remains unchanged, because it must have been the shortest path value

CS 477/677 - Lecture 20 42

Shortest Path Properties

• Path relaxation propertyLet p = v0, v1, . . . , vk be a shortest path from s

= v0 to vk. If we relax, in order, (v0, v1), (v1, v2), . . . ,

(vk-1, vk), even intermixed with other relaxations,

then d[vk ] = δ(s, vk).

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v1 v2

113

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14d[v1] = δ(s, v1)

d[v2] = δ(s, v2)

d[v3] = δ(s, v3)

d[v4] = δ(s, v4)v3

v4

CS 477/677 - Lecture 20 43

Single-Source Shortest Paths in DAGs

• Given a weighted DAG: G = (V, E) – solve the shortest path problem

• Idea:– Topologically sort the vertices of the graph

– Relax the edges according to the order given by the topological sort

• for each vertex, we relax each edge that starts from that vertex

• Are shortest-paths well defined in a DAG?– Yes, (negative-weight) cycles cannot exist

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CS 477/677 - Lecture 20 44

Dijkstra’s Algorithm

• Single-source shortest path problem:– No negative-weight edges: w(u, v) > 0 (u, v) E

• Maintains two sets of vertices:– S = vertices whose final shortest-path weights have

already been determined

– Q = vertices in V – S: min-priority queue• Keys in Q are estimates of shortest-path weights (d[v])

• Repeatedly select a vertex u V – S, with the minimum shortest-path estimate d[v]

CS 477/677 - Lecture 20 45

Dijkstra (G, w, s)

1. INITIALIZE-SINGLE-SOURCE(V, s)

2. S ←

3. Q ← V[G]

4. while Q

5. do u ← EXTRACT-MIN(Q)

6. S ← S {u}

7. for each vertex v Adj[u]8. do RELAX(u, v, w)

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CS 477/677 - Lecture 20 46

Example

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CS 477/677 - Lecture 20 47

Dijkstra (G, w, s)1. INITIALIZE-SINGLE-SOURCE(V, s)

2. S ←

3. Q ← V[G]

4. while Q

5. do u ← EXTRACT-MIN(Q)

6. S ← S {u}

7. for each vertex v Adj[u]8. do RELAX(u, v, w)

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