graph theory ch. 2. trees and distance 1 chapter 2 trees and distance 2.1 basic properties 2.2...

Graph Theory

Ch. 2. Trees and Distance1

Chapter 2

Trees and Distance

2.1 Basic properties

2.2 Spanning tree and enumeration

2.3 Optimization and trees

Graph Theory


Acyclic, tree2.1.1

A graph with no cycle is acyclic A forest is an acyclic graph A tree is a connected acyclic graph A leaf (or pendant vertex ) is a vertex of degree 1

tree tree

forest

leaf

Graph Theory


Spanning Subgraph 2.1.1

A spanning subgraph of G is a subgraph with vertex set V(G)

A spanning tree is a spanning subgraph that is a tree

Spanning subgraph Spanning tree

Graph Theory


Lemma. Every tree with at least two vertices has at least two leaves. 2.1.3

Proof: (1/2)

– A connected graph with at least two vertices has an edge.

– In an acyclic graph, an endpoint of a maximal nontrivial path has no neighbor other than its neighbor on the path.

– Hence the endpoints of such a path are leaves.

Maximal path

Impossible!

Cycle occurs

Impossible!

It is Maximal.

Graph Theory


Proof: (2/2)– Let v be a leaf of a tree G, and that G’=G-v.

– A vertex of degree 1 belongs to no path connecting two other vertices.

– Therefore, for u, w V(G’), every u, w-path in G is also in G’.

– Hence G’ is connected.

– Since deleting a vertex cannot create a cycle, G’ also is acyclic.

– Thus G’ is a tree with n-1 vertices.

Lemma. Deleting a leaf from a n-vertex tree produces a tree with n-1 vertices. 2.1.3

Graph Theory


For u, w V(G’ ), every u, w-path in G is also in G’.

u w

G

G’

v

Graph Theory


Theorem 2.1.4. For an n-vertex graph G (with n1), the following are equivalent (and characterize the trees with n vertices) 2.1.4

A) G is connected and has no cyclesB) G is connected and has n-1 edgesC) G has n-1 edges and no cyclesD) For u, vV(G), G has exactly one u, v-path

Proof: We first demonstrate the equivalence of A, B, and C by proving that any two of {connected, acyclic, n-1 edges} together imply the third

Graph Theory


A{B,C}. connected, acyclic n-1 edges

Theorem 2.1.4 Continue We use induction on n. For n=1, an acyclic 1-vertex graph has no edge For n >1, we suppose that implication holds

for graphs with fewer than n vertices – Given an acyclic connected graph G, Lemma

2.1.3 provides a leaf v and states that G’=G-v also is acyclic and connected (see figure in previous proof)

– Applying the induction hypothesis to G’ yields e(G’)=n-2

– Since only one edge is incident to v, we have e(G)=n-1

Graph Theory


B{A, C} Connected and n-1 edges acyclic

Theorem 2.1.4 continue

If G is not acyclic, delete edges from cycles of G one by one until the resulting graph G’ is acyclic

Since no edge of a cycle is a cut-edge(Theorem 1.2.14), G’ is connected – Now the preceding paragraph implies that

e(G’) = n-1

– Since we are given e(G)=n-1, no edges were deleted

Thus G’=G, and G is acyclic

Graph Theory


C{A, B} n-1 edges and no cycles connected

Theorem 2.1.4 continue

Let G1,…,Gk be the components of G

Since every vertex appears in one component, in(Gi)=n

– Since G has no cycles, each component satisfies property A. Thus e(Gi) = n(Gi) - 1

– Summing over i yields e(G)=I[n(Gi)-1]= n-k

We are given e(G)=n-1,so k =1, and G is connected

Graph Theory


AD Connected and no cycles For u, vV(G), one and only one u, v-path exists

Theorem 2.1.4

Since G is connected, each pair of vertices is connected by a path

If some pair is connected by more than one , we choose a shortest (total length) pair P, Q of distinct paths with the same endpoints

By this extremal choice, no internal vertex of P or Q can belong to the other path

This implies that PQ is a cycle, which contradicts the hypothesis Ap

vuq

DA For u, vV(G), one and only one u, v-path exists connected and no cycles Theorem

2.1.4

If there is a u,v-path for every u,vV(G), then G is connected

If G has a cycle C, then G has two u,v-paths for u,v V(G); which contradicts the hypothesis D

Hence G is acyclic (this also forbids loops).

Graph Theory


Graph Theory


Corollary: Every edge of a tree is a cut-edge 2.1.5

Proof:

A tree has no cycles Theorem 1.2.14 implies that every edge is

a cut-edge.

Graph Theory


Corollary: Adding one edge to a tree forms exactly one cycle 2.1.5

Proof:

A tree has a unique path linking each pair of vertices (Theorem2.1.4D)

Joining two vertices by an edge creates exactly one cycle

Graph Theory


Corollary: Every connected graph contains a spanning tree 2.1.5

Proof:

Similar to the proof of BA, C in Theorem 2.1.4 Iteratively deleting edges from cycles in a

connected graph yields a connected acyclic subgraph

Graph Theory


Proposition: If T, T’ are spanning trees of a connected graph G and eE(T)-E(T’), then there

is an edge e’E(T’)- E(T) such that T-e+e’ is a spanning tree of G. 2.1.6

T’T T-e+e’

ee’

G

Graph Theory


Proposition: If T, T’ are spanning trees of a connected graph G and eE(T)-E(T’), then there is an edge e’E(T’)- E(T) such

that T-e+e’ is a spanning tree of G. 2.1.6Proof:

Every edge of T is a cut-edge of T. Let U and U’ be the two components of T-e.

Since T’ is connected, T’ has an edge e’ with endpoints in U and U’.

Now T-e+e’ is connected, has n(G)-1 edges, and is a spanning tree of G.

T’T T-e+e’e

e’

Graph Theory


Distance in trees and Graphs

Assume G has a u, v-path

Then the distance from u to v, written dG(u,v) or d(u,v),

is the least length of a u,v-path. If G has no such path, then d(u,v)=

Graph Theory


Distance in trees and Graphs

The diameter (diam G) is maxu,vV(G) d(u,v)

– Upper bound of distance between every pair

The eccentricity of a vertex u is

(u) = maxvV(G) d(u,v)

– Upper bound of the distance from u to the others

The radius of a graph G is rad G = minuV(G) (u)

– Lower bound of the eccentricity

Graph Theory


Distance, Diameter, Eccentricity, and Radius

a b

c

d

e

f

g

Distance(f,c) : 2Distance(g,c): 2Distance(a,c): 3

Eccentricity(f):2Eccentricity(a): 3

Diameter: 3 Radius: 2

Graph Theory


If G is a simple graph, then diam G 3 diam 3 2.1.11

Proof: 1/3

Since diam G > 2, there exist nonadjacent vertices u, vV(G) with no common neighbor – If any pair of nonadjacent vertices has a

common neighbor, the distance of every pair is less than or equal to 2 and diam G = 2

G

u v

Not every pair is of this kind

A pair (u, v) of this kind exists

Graph Theory



Proof: 2/3

Hence every xV( ) - {u,v} has at least one of {u,v} as a nonneighbor– Either u or v is not connected to x

Equivalently, this makes x adjacent to at least one of {u,v} in G

u vG

G

Everyone of these vertices has at least one of {u,v} as a nonneighbor

G

Graph Theory



Proof: 3/3

Since also u v E( ), for every pair x, y there is an x, y-path of length at most 3 in through {u,v}. Hence diam 3

u vG

G

GG

G

Graph Theory


Center 2.1.12

Definition: The center of a graph G is the subgraph induced by the vertices of minimum eccentricity.

The center of a graph is the full graph if and only if the radius and diameter are equal.

Center

Recall: The eccentricity of a vertex u is the upper bound of the distance from u to the others

Graph Theory


The center of a tree is a vertex or an edge 2.1.13

Proof: We use induction on the number of vertices in a tree T.

Basis step: n(T)≦ 2. With at most two vertices, the center is the entire tree.

Graph Theory


Theorem. 2.1.13 Continue Induction step: n(T)>2.

– Let T’ = T- {leaves}. By Lemma 2.1.3, T’ is a tree. – Since the internal vertices on the paths between

leaves of T remain, T’ has at least one vertex.– Every vertex at maximum distance in T from a vertex

uV(T) is a leaf (otherwise, the path reaching it from u can be extended farther).

T = Green PinkT’ = Green

u

The max path from u

The end must be a leaf

Graph Theory


Theorem. 2.1.13 Continue– Since all the leaves have been removed and no path

between two other vertices uses a leave, T’(u)= T(u)-1 for every uV(T’).

– Also, the eccentricity of a leaf in T is greater than the eccentricity of its neighbor in T.

– Hence the vertices minimizing T(u) are the same as the vertices minimizing T’(u).

uT’(u)= T(u)-1

{v| v has min (v) in T} ={v’| v’ has min (v’) in T’}

Graph Theory


Theorem. 2.1.13 Continue It is shown T and T’ have the same center. By the induction hypothesis, the center of T’ is a

vertex or an edge.

T T’ T”

Graph Theory


Spanning Trees and Enumeration 2.2

There are 2c(n,2) simple graphs with vertex set [n]={1,…,n}.– since each pair may or may not form an

edge.

How many of these are trees?

Max number of edges = c(n,2)

Can either appear or disappear

Graph Theory


Enumeration of Trees 2.2

One or two vertices, then there is only one tree.

Three vertices, three trees. Four vertices, then four stars and 12 paths,

total 16. Five vertices, then there are 125 trees.

Graph Theory


Algorithm for tree generation by using Prűfer code 2.2.1

Algorithm. (Prűfer code) Production of f(T)=(a1,…,an-2)

Input: A tree T with vertex set S N

Iteration: At the i th step, delete the least remaining leaf, and say that ai is the neighbor of the deleted leaf

2 7 1 4 3

6 8 5

1. Delete 2 a1= 72. Delete 3 a2= 43. Delete 5 a3= 44. Delete 4 a4= 15. Delete 6 a5= 76. Delete 7 a6= 1

Graph Theory


Theorem: For a set SN of size n, there are nn-2 trees with vertex set S 2.2.3

Proof: (sketch) Trees with vertex set S Sn-2 of lists of

length n-2

– One prufer code one tree.

– There are nn-2 codes.

– Proved by induction

Graph Theory


Spanning Trees in Graphs 2.2.6

Example. A kite. To count the spanning trees: Four are path around the outside cycle in the drawing The remaining spanning trees use the diagonal edge

– Since we must include an edge to each vertex of degree 2, we obtain four more spanning trees.

The total is eight.

Graph Theory


Contraction 2.2.7

In a graph G, contraction of edge e with endpoints u, v is the replacement of u and v with a single vertex whose incident edges are the edges other than e that were incident to u or v

The resulting graph G·e has one less edge than G

u

e

vG G·e

Graph Theory


Proposition. Let (G) denote the number of spanning trees of a graph G. If eE(G) is not a loop, then (G)= (G-e)+ (G·e) 2.2.8

(G-e): The number of trees without e (G·e): The number of trees with e A spanning tree in G.e A spanning tree having e

in G

e

G

G·e

G-e

=

Graph Theory


Proposition. Let (G) denote the number of spanning trees of a graph G. If eE(G) is not a loop, then (G)= (G-e)+ (G·e) 2.2.8

Proof: 1/2

The spanning trees of G that omit e are precisely the spanning trees of G-e

Need to show that G has (G·e) spanning trees containing e– It must be shown that contraction of e

defines a bijection from the set of spanning trees of G containing e to the set of spanning trees of G·e

Graph Theory


Proposition. Let (G) denote the number of spanning trees of a graph G. If eE(G) is not a loop, then (G)= (G-e)+ (G·e)

2.2.8

Proof:2/2

When contract e in a spanning tree having e, obtain a spanning tree of G·e because

– The resulting subgraph of G·e is spanning and connected

– It has the right number of edges

– The other edges maintain their identity under contraction

So no two trees are mapped to the same spanning tree of G·e by this operation.

Graph Theory


A Matrix Tree computation. 2.2.12

Given a loopless graph G with vertex set v1, …., vn

Let aij be the number of edges with endpoints vi and vj

Let Q be the matrix in which – entry (i, j) is –ai,j when i j

and is d(vi) when i=j

Let Q* is a matrix obtained by deleting row s and column t of Q, then

(G) = (-1)s+t detQ*

Graph Theory



Theorem 2.2.12 instructs us– Form a matrix by putting the vertex degrees on

the diagonal and subtracting the adjacency matrix

– Delete a row and a column and take the eterminant

Graph Theory



Consider the kite of Example 2.2.9, – The vertex degrees are 3,3,2,2

– We form the matrix on the left below and then

– We take the determinant of the matrix in the middle

– The result is the number of spanning trees

Graph Theory


Example 2.2.11

2011

0211

1131

1113

201

021

113

8

v1

v2

v3

v4

v1 v2 v3 v4v1 v2 v3 v4

Graph Theory


Minimum Spanning Tree 2.3

In a connected weighted graph of possible communication links, all spanning trees have n-1 edges; we seek one that minimizes or maximizes the sum of the edge weights.

– Kruskal’s Algorithm

– Prim’s Algorithm

Graph Theory


Kruskal’s Algorithm for Minimum Spanning Tree 2.3.1

Input: A weighted connected graph Idea:

– Maintain an acyclic spanning subgraph H.

– Enlarging it by edges with low weight to form a spanning tree.

– Consider edges in nondecreasing order of weight.

Graph Theory


Kruskal’s Algorithm for Minimum Spanning Tree 2.3.1

Initialization: Set E(H)=. Iteration: If the next cheapest edge joins

two components of H, then include it; otherwise, discard it. Terminate when H is connected.

HJoin Two vertices in one component. Cycle occurs.

Not Allowed!

Join two components. It works

Graph Theory


Example of using Kruskal’s Algorithm

1

9

4

3

12

11

7

8 2

6

510

1

9

4

3

12

11

7

8 2

6

510

1

9

4

3

12

11

7

8 2

6

510

1

9

4

3

12

11

7

8 2

6

510

1

9

4

312

11

78 2

6

510

1

9

4

3

12

11

78 2

6

510

Graph Theory


Theorem: In a connected weighted graph G, Kruskal’s Algorithm constructs a minimum-weight spanning tree. 2.3.3

Proof: 1/3

We show first that the algorithm produces a tree

We never choose an edge that completes a cycle

If the final graph has more than one component, then there is no edge joining two of them and G is not connected

– Since G is connected, some such edge exists and we considered it.

Thus the final graph is connected and acyclic, which makes it a tree.

Graph Theory


Theorem 2.3.3 Continue

Proof: continue Let T be the resulting tree, and let T* be a spannig tree of

minimum weight. If T=T*, we are done. If TT*, let e be the first edge chosen for T that is not in

T*. Adding e to T* creates one cycle C. Since T has no cycle, C has an edge e’E(T). Consider the spanning tree T*+e-e’

T: …………. e, …………T*: …………. e*,…………

Identical

The first edge chosen for T that is not in T*

Graph Theory



Proof: continue Since T* contains e’ and all the edges of T chosen

before e, both e’ and e are available when the algorithm chooses e, and hence w(e)w(e’)

Thus T*+e-e’ is a spanning tree with weight at most T* that agrees with T for a longer initial list of edges than T* does.

T: …………. e, ………… T*: …………. e*,…………

Identical

T: …………… e, ………… T*+e-e’: …………… e, …………

Identical

Graph Theory



Proof: continue Repeating this argument eventually yields a minimum-

weight spanning tree that agrees completely with T. Phrased extremely, we have proved that the minimum

spanning tree agreeing with T the longest is T itself.

Graph Theory


Shortest Paths

How can we find the shortest route from one location to another?

Graph Theory


Dijkstra’s Algorithm2.3.5

Input: A graph (or digraph) with nonnegative edge weights and a starting vertex u. – The weight of edge xy is w(xy)

– Let w(xy)= if xy is not an edge

u

a d

e

cb

1

5

2

6

44

5

3

Graph Theory



Idea: – Maintain the set S of vertices to which a

shortest path from u is known

– Enlarge S to include all vertices. • To do this, maintain a tentative distance t(z)

from u to each zS, being the length of the shortest u, v-path yet found.

u

S

u

S

The one nearest to u

v

Graph Theory



Initialization: Set S={u}; t(u)=0; t(z)=w(uz) for zu

Iteration:

1. Select a vertex v outside S such that t(v)=minzS t(z).

2. Add v to S.

3. Explore edges from v to update tentative distance: for each edge vz with zS, update t(z) to min{t(z), t(v)+w(vz)}

The iteration continues until S=V(G) or until t(z)= for every zS. At the end, set d(u, v)=t(v) for all v.

Graph Theory


Tentative distance v.s. Shortest distance 2.3.5

The tentative distance of a vertex may not be the shortest distance, for example:

– t(d ) = 6 and not the minimum

– Actually d(d ) = 5

u

ad

e

b

1

5

2

6

2

3d=0

d=1

t=3

t=6

u

ad

e

b

1

5

2

6

2

3

d=0

d=1t=5

d=3

t=9

Graph Theory


Example: Find Shortest paths by using Dijkstra’s Algorithm 2.3.6

u

a d

e

cb

15

2

6

44

53

u

a d

e

cb

15

2

6

44

5

3d=0

t=1

t=3

Graph Theory


Example: Find Shortest paths by using Dijkstra’s Algorithm continue 2.3.6

u

ad

e

c

b

1

5

2

6

4

45

3d=0

d=1

t=3

t=6

t=5

u

a

d

e

c

b

1 5 2

6

4

4

53

d=0

d=1

d=3 t=6

t=5

Graph Theory


Example: Find Shortest paths by using Dijkstra’s Algorithm continue 2.3.6

u

a

d

e

c

b

15

2

64

4

5

3

d=0

d=1

d=3

t=11

d=5

t=6

u

a

d

e

c

b

15

2

64

4

5

3

d=0

d=1

d=3

t=8

d=5

d=6

Graph Theory


Theorem: Given a (di)graph G and a vertex uV(G), Dijkstra’s algorithm compute d(u, z) for every z V(G) 2.3.7

Proof: We prove the stronger statement that at

each iteration,1) For zS, t(z)=d(u, z), and 2) For zS, t(z) is the least length of a u, z-path

reaching z directly from S.

Graph Theory



We use induction on k=|S|. Basis step: k=1 From the initialization,

– S={u}, d(u, u)=t(u)=0,– the least length of a u, z-path reaching z

directly from S is t(z)=w(u,z), which is infinite when uz is not an edge

Graph Theory



Induction step: Suppose that when |S|=k, (1) and (2) are true. – Let v be a vertex among zS such that t(z) is smallest.

– The algorithm now choose v; let S’=S{v}. We first argue that d(u, v)=t(v). A shortest u, v-path must exit S before reaching v. The induction hypothesis states that the length of the shortest path going directly to v from S is t(v). The induction hypothesis and choice of v also guarantee that a path visiting any vertex outside S and later reaching v has length at least t(v).

– Hence d(u, v)=t(v), and (1) holds for S’.

Graph Theory



To prove (2) for S’, let z be a vertex outside S other than v.

– By the hypothesis, the shortest u, v-path reaching z directly from S has length t(z) ( if there is no such path).

– When we add v to S, we must also consider paths reaching z from v. Since we have now computed d(u, v)=t(v), the shortest such path has length t(v)+w(vz), and we compare this with the previous value of t(z) to find the shortest path reaching z directly from S’.

We have verified that (1) and (2) hold for the new set S’ of size k+1; this completes the induction step.

Graph Theory


Prim’s Algorithmfor Minimum Spanning Tree

Input: A graph (or digraph) with nonnegtive edge weights and a starting vertex u. The weight of edge xy is w(xy); let w(xy)= if xy is not an edge.

Idea: Maintain a tree T including u at the beginning, enlarging T to include all vertices. To do this, select the cheapest edge from the edges E={(x,v) |x T, vT }.

Graph Theory


Prim’s Algorithm

Initialization: Set T={u}.

Iteration:

– Select a vertex v outside T such that w(x,v)=minx T, vT w(x,v)

– Add v to T

– The iteration continues until V(T )=V(G).

Graph Theory


Dijkstra’s Algorithm v.s. Prim’s Algorithm

u

ad

ec

bd=0

d=1

t=3

t=6

t=5

The vertex which is the nearest to u

8

The end vertex of the cheapest

edge

3

6

Graph Theory


Algorithm - Breadth First Search 2.3.8

Input: An unweighted graph (or digraph) and a start vertex u.

Idea:

Maintain a set R of vertices that have been reached but not searched and a set S of vertices that have been searched

The set R is maintained as a First-In First-Out list (queue), so the first vertices found are the first vertices explored

Graph Theory


Algorithm - Breadth First Search 2.3.8

Initialization: R={u}, S=, d(u, u)=0

Iteration:

1. As long as R, we search from the first vertex v of R

2. The neighbors of v not in SR are added to the back of R and assigned distance d(u, v)+1

3. Then v is removed from the front of R and placed in S

Graph Theory


Example of BFS

a

bc d

e f g h i j

Order of BFS: a, b, c, d, e, f, g, h, i, j, …

a, c, b, d, f, h, e, g, i, j, …

b, a, g, f, e, c, d, … h, i, j, …

Graph Theory


Breadth-F-S vs. Depth-F-S

a

bc d

e f g h i j

a

bc d

e f g h i j

a,b,c,d,e,f,g,h,i,j, … a,b,e,f, …c, h,…

Graph Theory


Application: Chinese Postman Problem 2.3.9

A mail carrier must traverse all edges in a road network, starting and ending at the Post Office.

– The edges has nonnegative weights representing distance or time.

– We seek a closed walk of minimum total length that uses all the edges

– This is the Chinese Postman Problem,• named in honor of the Chinese mathematician

Guan Meigu [1962], who proposed it.

Graph Theory


Application: Chinese Postman Problem continue 2.3.9

If every vertex is even, then the graph is Eulerian and the answer is the sum of the edge weights.

Otherwise, we must repeat edges. Every traversal is an Eulerian circuit of a graph obtained by duplicating edges.

Finding the shortest traversal is equivalent to finding the minimum total weight of edges whose duplication will make all vertex degrees even .

Graph Theory



We say “duplication” because we need not use an edge more than twice

If we use an edge three or more times in making all vertices even, then deleting two of those copies will leave all vertices even

There may be many ways to choose the duplicated edges

Graph Theory



If there are only two odd vertices, then – We can use Dijkstra’s Algorithm to find the

shortest path between them and solve the problem

If there are 2k odd vertices, then – Use Dijkstra’s algorithm to find the shortest paths

connecting each pair of odd vertices

– Use these lengths as weights on the edges of K2k

– Find the minimum total weight of k edges that pair up these 2k vertices. This is a maximum matching problem.

Graph Theory



• A vertex: An odd vertex • An edge between u and v: The shortest path between u and v in the given graph

16

20

40

15

20

10A matching in a general graph with minimum total weight

Graph Theory


Rooted Tree 2.3.11

A rooted tree is a tree with one vertex r chosen as root.

For each vertex v, let P(v) be the unique v, r-path.

– The parent of v is its neighbor on P(v);

– Its children are its other neighbors.

– Its ancestors are the vertices of P(v)-v.

– Its descendants are the vertices u such that P(u) contains v.

– The leaves are the vertices with no children.

– A rooted plane tree or planted tree is a rooted tree with a left-to–right ordering specified for the children of each vertex.

Graph Theory


Rooted Tree 2.3.11

root

v

Parent of v,Ancester of v

A leaf,A child of v

Binary Tree 2.3.12

A binary tree is a rooted plane tree where each vertex has at most two children, and each child of a vertex is designated as its left child or right child

The subtrees rooted at the children of the root are the left subtree and the right subtree of the tree

A k-ary tree allows each vertex up to k children

Graph Theory


Graph Theory


Huffman’s Algorithm 2.3.13

Input: Weights (frequencies or probabilities) p1,…,pn

Output: Prefix-free code (equivalently, a binary tree)

Idea: Infrequent items should have longer codes; put infrequent items deeper by combining them into parent nodes

Graph Theory



Initial case: When n=2, the optimal length is one, with 0 and 1 being the codes assigned to the two items (the tree has a root and two leaves; n=1 can also be used as the initial case)

Graph Theory



Recursion:

Replace the two least likely items p, p’ with a single item q of weight p+p’ when n>2

Treat the smaller set as a problem with n-1 items.

After solving it, give children with weights p, p’ to the resulting leaf with weight q. – Equivalently, replace the code computed for

the combined item with its extensions by 1 and 0, assigned to the items that were replaced.

Graph Theory


Example of Huffman coding 2.3.14

Consider eight items with frequencies 5, 1, 1, 7, 8, 2, 3, 6

Algorithm 2.3.13 combines items according to the tree on the left below, working from the bottom up

– First the two items of weight 1 combine to from one of weight 2

2

5 1 1 7 8 2 3 6

Graph Theory


Huffman coding 2.3.14

42

5 1 1 7 8 2 3 6

– Now this and the original item of weight 2 are the least likely and combine to form an item of weight 4.

Graph Theory



2

5 1 1 7 8 2 3 6

42

5 1 1 7 8 2 3 6

7

4

2

5 1 1 7 8 2 3 6

7

114

2

5 1 1 7 8 2 3 6

Graph Theory



33

19 14

7

11 4

2

5 1 1 7 8 2 3 6

6:101

8:11

5:100

7:01

3:001

2:0001

1:000011:00000

0 1

0

0

0

1

1

1

0 1

0

1

10

Graph Theory


Theorem 2.3.15

Given a probability distribution {pi} on n items, Huffman’s Algorithm produces the prefix-free code with minimum excepted length

graph theory ch. 2. trees and distance 1 chapter 2 trees and distance 2.1 basic properties 2.2...

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tree g

nvertex graph g

acyclic connected graph

acyclic n

cycles of g

resulting graph g

g yields

components of g