5/5 - (1 vote)

Data Structures and Algorithms Spring 2022
Instructor: Chunhao Wang
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 1 / 12

Greedy algorithms

Greedy algorithms Minimum Spanning Tree

Running time of Kruskals algorithm (I)
Depends on how we implement make set, find set, and union
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 2 / 12

Running time of Kruskals algorithm (I)
Depends on how we implement make set, find set, and union Using linked list:
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 2 / 12

Running time of Kruskals algorithm (I)
Depends on how we implement make set, find set, and union Using linked list:
Cost of union:
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 2 / 12

Running time of Kruskals algorithm (I)
Depends on how we implement make set, find set, and union Using linked list:
Cost of union: O(length of the shorter list)
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 2 / 12

Running time of Kruskals algorithm (I)
Depends on how we implement make set, find set, and union Using linked list:
Cost of union: O(length of the shorter list) Using an array to implement it:
vertex head
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 2 / 12

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v)
Sort E in increasing order of edge weights for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 3 / 12

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v) ;
Sort E in increasing order of edge weights for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
Chunhao Wang CMPSC 465 Spring 2022
Mar 17, 2022

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v) ;
Sort E in increasing order of edge weights ; for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
Worst-case cost for union: O(|V |).
// O(|V|) // O(|E|log|V|)
Chunhao Wang CMPSC 465 Spring 2022
Mar 17, 2022

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v) ;
Sort E in increasing order of edge weights ; for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
// O(|V|) // O(|E|log|V|)
Worst-case cost for union: O(|V|). What about the cost for lines 6-9? Consider a single v V . Once its touched in some union operation, the size of the set at least doubles. Since the maximum size of a set can be |V |, each v is touched at most O(log |V |) times
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 3 / 12

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v) ;
Sort E in increasing order of edge weights ; for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
// O(|V|) // O(|E|log|V|)
Worst-case cost for union: O(|V|). What about the cost for lines 6-9? Consider a single v V . Once its touched in some union operation, the size of the set at least doubles. Since the maximum size of a set can be |V |, each v is touched at most O(log |V |) times
At most |V | vertices are involved in union operations, so the total cost of lines 6-9:
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 3 / 12

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v) ;
Sort E in increasing order of edge weights ; for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
// O(|V|) // O(|E|log|V|)
Worst-case cost for union: O(|V|). What about the cost for lines 6-9? Consider a single v V . Once its touched in some union operation, the size of the set at least doubles. Since the maximum size of a set can be |V |, each v is touched at most O(log |V |) times
At most |V | vertices are involved in union operations, so the total cost of lines 6-9: O(|V | log |V |)
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 3 / 12

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v) ;
Sort E in increasing order of edge weights ; for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
// O(|V|) // O(|E|log|V|)
Worst-case cost for union: O(|V|). What about the cost for lines 6-9? Consider a single v V . Once its touched in some union operation, the size of the set at least doubles. Since the maximum size of a set can be |V |, each v is touched at most O(log |V |) times
At most |V | vertices are involved in union operations, so the total cost of
lines 6-9: O(|V | log |V |)
Total cost of the algorithm:
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 3 / 12

Running time of Kruskals algorithm (II)
def Kruskal MST(undirected G = (V , E ), weights w = (we )eE ): Set A := { };
for v V :
make set(v) ;
Sort E in increasing order of edge weights ; for (u,v) E:
if find set(u) = find set(v): A := A {(u, v)};
union(u, v );
// O(|V|) // O(|E|log|V|)
Worst-case cost for union: O(|V|). What about the cost for lines 6-9? Consider a single v V . Once its touched in some union operation, the size of the set at least doubles. Since the maximum size of a set can be |V |, each v is touched at most O(log |V |) times
At most |V | vertices are involved in union operations, so the total cost of
lines 6-9: O(|V | log |V |)
Total cost of the algorithm: O(|E|log|V|)
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 3 / 12

Alternative data structure
The linked-list implementation is good enough, but there exist better data structures to improve the worst-case cost for union
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 4 / 12

Alternative data structure
The linked-list implementation is good enough, but there exist better data structures to improve the worst-case cost for union
Directed tree disjoint set:
Definition
(x): parent of x
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 4 / 12

Alternative data structure
The linked-list implementation is good enough, but there exist better data structures to improve the worst-case cost for union
Directed tree disjoint set:
Definition
(x): parent of x
root node: x s.t. (x) = x
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 4 / 12

Alternative data structure
The linked-list implementation is good enough, but there exist better data structures to improve the worst-case cost for union
Directed tree disjoint set:
Definition
(x): parent of x
root node: x s.t. (x) = x
rank(x): number of the edges in the longest simple path from x to a leaf
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 4 / 12

Operations of direct tree disjoint set (I)
make set(v)
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 5 / 12

Operations of direct tree disjoint set (I)
make set(v)
def make set(v): (v) := v;
rank(v) = 0;
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 5 / 12

Operations of direct tree disjoint set (I)
make set(v)
def make set(v): (v) := v;
rank(v) = 0; Cost: O(1)
Chunhao Wang
CMPSC 465 Spring 2022
Mar 17, 2022

Operations of direct tree disjoint set (I)
make set(v)
def make set(v): (v) := v;
rank(v) = 0; Cost: O(1)
find set(v)
Chunhao Wang
CMPSC 465 Spring 2022
Mar 17, 2022

Operations of direct tree disjoint set (I)
make set(v)
def make set(v): (v) := v;
rank(v) = 0;
Cost: O(1) find set(v)
def find set(v): while v = (v):
v := (v); return v;
Chunhao Wang
CMPSC 465 Spring 2022
Mar 17, 2022

Operations of direct tree disjoint set (I)
make set(v)
def make set(v): (v) := v;
rank(v) = 0;
Cost: O(1) find set(v)
def find set(v): while v = (v):
v := (v); return v;
Cost: O(depth of the node in the tree)
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 5 / 12

Operations of direct tree disjoint set (I)
make set(v)
def make set(v): (v) := v;
rank(v) = 0;
Cost: O(1) find set(v)
def find set(v): while v = (v):
v := (v); return v;
Cost: O(depth of the node in the tree) what about union?
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 5 / 12

Operations of direct tree disjoint set (II)
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 6 / 12

Operations of direct tree disjoint set (II)
Option 1 Option 2
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 6 / 12

Operations of direct tree disjoint set (II)
Option 1 Option 2
Basic idea: attach the smaller ranked tree to a larger one
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 6 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y);
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y); if rank(rx ) > rank(ry ):
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y); if rank(rx ) > rank(ry ):
(ry) := rx;
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y); if rank(rx ) > rank(ry ):
(ry) := rx; else:
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y); if rank(rx ) > rank(ry ):
(ry) := rx; else:
(rx) := ry;
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y); if rank(rx ) > rank(ry ):
(ry) := rx;
(rx) := ry;
if rank(rx ) == rank(ry ):
Chunhao Wang
CMPSC 465 Spring 2022
Mar 17, 2022

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y); if rank(rx ) > rank(ry ):
(ry) := rx;
(rx) := ry;
if rank(rx ) == rank(ry ):
rank(ry ) := rank(ry ) + 1;
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Operations of direct tree disjoint set (II)
def union(x,y):
rx := find set(x), ry := find set(y); if rank(rx ) > rank(ry ):
(ry) := rx;
(rx) := ry;
if rank(rx ) == rank(ry ):
rank(ry ) := rank(ry ) + 1;
Cost: dominated by find set
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 7 / 12

Cost of find set using directed tree disjoint set
Observation
Root note with rank k is formed by the merge of two rank k 1 trees
Chunhao Wang CMPSC 465 Spring 2022 Mar 17, 2022 8 / 12

Cost of find set using directed tree di

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