David Luebke 1 6/1/2014 CS 332: Algorithms Medians and Order Statistics Structures for Dynamic Sets.
CS 332: Algorithms
description
Transcript of CS 332: Algorithms
David Luebke 1 04/21/23
CS 332: Algorithms
Dijkstra’s Algorithm Continued
Disjoint-Set Union
Return to MST (Kruskal)
Amortized Analysis
David Luebke 2 04/21/23
Review: Minimum Spanning Tree
Problem: given a connected, undirected, weighted graph, find a spanning tree using edges that minimize the total weight
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David Luebke 3 04/21/23
Review: Minimum Spanning Tree
MSTs satisfy the optimal substructure property: an optimal tree is composed of optimal subtrees
If T is MST of G, and A T is a subtree of T, and (u,v) is the min-weight edge connecting A to V-A, then (u,v) T
David Luebke 4 04/21/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
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David Luebke 5 04/21/23
Review: Prim’s Algorithm
MST-Prim(G, w, r)
Q = V[G];
for each u Q key[u] = ; key[r] = 0;
p[r] = NULL;
while (Q not empty)
u = ExtractMin(Q);
for each v Adj[u] if (v Q and w(u,v) < key[v]) p[v] = u;
key[v] = w(u,v);
What will be the running time?A: Depends on queue binary heap: O(E lg V) Fibonacci heap: O(V lg V + E)
David Luebke 6 04/21/23
Review: Single-Source Shortest Path
Problem: given a weighted directed graph G, find the minimum-weight path from a given source vertex s to another vertex v “Shortest-path” = minimum weight Weight of path is sum of edges E.g., a road map: what is the shortest path from
Chapel Hill to Charlottesville?
David Luebke 7 04/21/23
Review: Shortest Path Properties
Optimal substructure: the shortest path consists of shortest subpaths
Let (u,v) be the weight of the shortest path from u to v. Shortest paths satisfy the triangle inequality: (u,v) (u,x) + (x,v)
In graphs with negative weight cycles, some shortest paths will not exist
David Luebke 8 04/21/23
Review: Relaxation
Key technique: relaxation Maintain upper bound d[v] on (s,v):Relax(u,v,w) {
if (d[v] > d[u]+w) then d[v]=d[u]+w;
}
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David Luebke 9 04/21/23
Review: Bellman-Ford Algorithm
BellmanFord()
for each v V d[v] = ; d[s] = 0;
for i=1 to |V|-1
for each edge (u,v) E Relax(u,v, w(u,v));
for each edge (u,v) E if (d[v] > d[u] + w(u,v))
return “no solution”;
Relax(u,v,w): if (d[v] > d[u]+w) then d[v]=d[u]+w
Initialize d[], whichwill converge to shortest-path value
Relaxation: Make |V|-1 passes, relaxing each edge
Test for solution:have we converged yet?Ie, negative cycle?
David Luebke 10 04/21/23
Review: Bellman-Ford Algorithm
BellmanFord()
for each v V d[v] = ; d[s] = 0;
for i=1 to |V|-1
for each edge (u,v) E Relax(u,v, w(u,v));
for each edge (u,v) E if (d[v] > d[u] + w(u,v))
return “no solution”;
Relax(u,v,w): if (d[v] > d[u]+w) then d[v]=d[u]+w
What will be the running time?
David Luebke 11 04/21/23
Review: Bellman-Ford
Running time: O(VE) Not so good for large dense graphs But a very practical algorithm in many ways
Note that order in which edges are processed affects how quickly it converges
David Luebke 12 04/21/23
Review: DAG Shortest Paths
Problem: finding shortest paths in DAG Bellman-Ford takes O(VE) time. Do better by using topological sort
If were lucky and processes vertices on each shortest path from left to right, would be done in one pass
Every path in a dag is subsequence of topologically sorted vertex order, so processing verts in that order, we will do each path in forward order (will never relax edges out of vert before doing all edges into vert).
Thus: just one pass. Running time: O(V+E)
David Luebke 13 04/21/23
Review: Dijkstra’s Algorithm
Dijkstra(G)
for each v V d[v] = ; d[s] = 0; S = ; Q = V; while (Q ) u = ExtractMin(Q);
S = S {u}; for each v u->Adj[] if (d[v] > d[u]+w(u,v))
d[v] = d[u]+w(u,v);RelaxationStepNote: this
is really a call to Q->DecreaseKey()
David Luebke 14 04/21/23
Review: Dijkstra’s Algorithm
Dijkstra(G)
for each v V d[v] = ; d[s] = 0; S = ; Q = V; while (Q ) u = ExtractMin(Q);
S = S {u}; for each v u->Adj[] if (d[v] > d[u]+w(u,v))
d[v] = d[u]+w(u,v);Running time: O(E lg V) using binary heap for QCan acheive O(V lg V + E) with Fibonacci heaps
David Luebke 15 04/21/23
Dijkstra’s Algorithm
Dijkstra(G)
for each v V d[v] = ; d[s] = 0; S = ; Q = V; while (Q ) u = ExtractMin(Q);
S = S {u}; for each v u->Adj[] if (d[v] > d[u]+w(u,v))
d[v] = d[u]+w(u,v);Correctness: we must show that when u is removed from Q, it has already converged
David Luebke 16 04/21/23
Correctness Of Dijkstra's Algorithm
Note that d[v] (s,v) v Let u be first vertex picked s.t. shorter path than d[u] d[u] > (s,u) Let y be first vertex V-S on actual shortest path from su d[y] = (s,y)
Because d[x] is set correctly for y's predecessor x S on the shortest path, and When we put x into S, we relaxed (x,y), giving d[y] the correct value
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David Luebke 17 04/21/23
Correctness Of Dijkstra's Algorithm
Note that d[v] (s,v) v Let u be first vertex picked s.t. shorter path than d[u] d[u] > (s,u) Let y be first vertex V-S on actual shortest path from su d[y] = (s,y) d[u] > (s,u)
= (s,y) + (y,u) (Why?)= d[y] + (y,u) d[y] But if d[u] > d[y], wouldn't have chosen u. Contradiction.
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David Luebke 18 04/21/23
Disjoint-Set Union Problem
Want a data structure to support disjoint sets Collection of disjoint sets S = {Si}, Si Sj =
Need to support following operations: MakeSet(x): S = S {{x}} Union(Si, Sj): S = S - {Si, Sj} {Si Sj}
FindSet(X): return Si S such that x Si
Before discussing implementation details, we look at example application: MSTs
David Luebke 19 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
David Luebke 20 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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David Luebke 21 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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David Luebke 22 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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David Luebke 23 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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David Luebke 24 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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5
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David Luebke 25 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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5
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David Luebke 26 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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9
1
5
13
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Run the algorithm:
David Luebke 27 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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9
1
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13
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Run the algorithm:
David Luebke 28 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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Run the algorithm:
David Luebke 29 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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5
13
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David Luebke 30 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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9
1
5
13
1725
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Run the algorithm:
David Luebke 31 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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5
13
1725
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Run the algorithm:
David Luebke 32 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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9
1
5
13
1725
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Run the algorithm:
David Luebke 33 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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9
1
5
13?
1725
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21
Run the algorithm:
David Luebke 34 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19
9
1
5
13
1725
148
21
Run the algorithm:
David Luebke 35 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
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9
1
5
13
1725
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Run the algorithm:
David Luebke 36 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19
9
1
5
13
1725
148
21
Run the algorithm:
David Luebke 37 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19
9
1
5
13
17?25
148
21
Run the algorithm:
David Luebke 38 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19?
9
1
5
13
1725
148
21
Run the algorithm:
David Luebke 39 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19
9
1
5
13
1725
148
21?
Run the algorithm:
David Luebke 40 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19
9
1
5
13
1725?
148
21
Run the algorithm:
David Luebke 41 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19
9
1
5
13
1725
148
21
Run the algorithm:
David Luebke 42 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
2 19
9
1
5
13
1725
148
21
Run the algorithm:
David Luebke 43 04/21/23
Correctness Of Kruskal’s Algorithm
Sketch of a proof that this algorithm produces an MST for T: Assume algorithm is wrong: result is not an MST Then algorithm adds a wrong edge at some point If it adds a wrong edge, there must be a lower weight
edge (cut and paste argument) But algorithm chooses lowest weight edge at each step.
Contradiction Again, important to be comfortable with cut and
paste arguments
David Luebke 44 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
What will affect the running time?
David Luebke 45 04/21/23
Kruskal’s Algorithm
Kruskal()
{
T = ; for each v V MakeSet(v);
sort E by increasing edge weight w
for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T {{u,v}}; Union(FindSet(u), FindSet(v));
}
What will affect the running time? 1 Sort
O(V) MakeSet() callsO(E) FindSet() callsO(V) Union() calls
(Exactly how many Union()s?)
David Luebke 46 04/21/23
Kruskal’s Algorithm: Running Time
To summarize: Sort edges: O(E lg E) O(V) MakeSet()’s O(E) FindSet()’s O(V) Union()’s
Upshot: Best disjoint-set union algorithm makes above 3
operations take O(E(E,V)), almost constant Overall thus O(E lg E), almost linear w/o sorting
David Luebke 47 04/21/23
Disjoint Set Union
So how do we implement disjoint-set union? Naïve implementation: use a linked list to
represent each set:
MakeSet(): ??? time FindSet(): ??? time Union(A,B): “copy” elements of A into B: ??? time
David Luebke 48 04/21/23
Disjoint Set Union
So how do we implement disjoint-set union? Naïve implementation: use a linked list to represent each
set:
MakeSet(): O(1) time FindSet(): O(1) time Union(A,B): “copy” elements of A into B: O(A) time
How long can a single Union() take? How long will n Union()’s take?
David Luebke 49 04/21/23
Disjoint Set Union: Analysis
Worst-case analysis: O(n2) time for n Union’sUnion(S1, S2) “copy” 1 element
Union(S2, S3) “copy” 2 elements
…
Union(Sn-1, Sn) “copy” n-1 elements
O(n2)
Improvement: always copy smaller into larger Why will this make things better? What is the worst-case time of Union()?
But now n Union’s take only O(n lg n) time!
David Luebke 50 04/21/23
Amortized Analysis of Disjoint Sets
Amortized analysis computes average times without using probability
With our new Union(), any individual element is copied at most lg n times when forming the complete set from 1-element sets Worst case: Each time copied, element in smaller set
1st time resulting set size 2
2nd time 4
…
(lg n)th time n
David Luebke 51 04/21/23
Amortized Analysis of Disjoint Sets
Since we have n elements each copied at most lg n times, n Union()’s takes O(n lg n) time
We say that each Union() takes O(lg n) amortized time Financial term: imagine paying $(lg n) per Union At first we are overpaying; initial Union $O(1) But we accumulate enough $ in bank to pay for later
expensive O(n) operation. Important: amount in bank never goes negative
David Luebke 52 04/21/23
Amortized Analysis
Book describes 3 views of amortized analysis in Chapter 18: Aggregate Accounting Potential
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Amortized Analysis: Aggregate Method
Aggregate method This is what we just did for Union() n operations take time T(n) Average cost of an operation = T(n)/n Not very precise
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Amortized Analysis: Accounting Method
Accounting method We have done this with graph algorithms Charge each operation an amortized cost
Usually just guess/invent this cost Amount not used stored in “bank” Later operations can used stored work Balance must not go negative
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Amortized Analysis:Potential Method
Potential method “Stored work” of accounting method is viewed as
“potential energy” Most flexible and powerful approach See book if interested; we won’t go into (or test)
David Luebke 56 04/21/23
Amortized Analysis Example: Dynamic Tables
Implementing a table (e.g., hash table) for dynamic data, want to make it small as possible
Problem: if too many items inserted, table may be too small
Idea: allocate more memory as needed
David Luebke 57 04/21/23
Dynamic Tables
1. Init table size m = 1
2. Insert elements until number n > m
3. Generate new table of size 2m
4. Reinsert old elements into new table
5. (back to step 2)What is the worst-case cost of an insert?One insert can be costly, but the total?
David Luebke 58 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1
David Luebke 59 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2
David Luebke 60 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2Insert(3) 4 1 + 2 3
David Luebke 61 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2Insert(3) 4 1 + 2 3Insert(4) 4 1 4
David Luebke 62 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2Insert(3) 4 1 + 2 3Insert(4) 4 1 4Insert(5) 8 1 + 4 5
David Luebke 63 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2Insert(3) 4 1 + 2 3Insert(4) 4 1 4Insert(5) 8 1 + 4 5Insert(6) 8 1 6
David Luebke 64 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2Insert(3) 4 1 + 2 3Insert(4) 4 1 4Insert(5) 8 1 + 4 5Insert(6) 8 1 6Insert(7) 8 1 7
David Luebke 65 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2Insert(3) 4 1 + 2 3Insert(4) 4 1 4Insert(5) 8 1 + 4 5Insert(6) 8 1 6Insert(7) 8 1 7Insert(8) 8 1 8
David Luebke 66 04/21/23
Analysis Of Dynamic Tables
Let ci = cost of ith insert
ci = i if i-1 is exact power of 2, 1 otherwise
Example: Operation Table Size Cost
Insert(1) 1 1 1Insert(2) 2 1 + 1 2Insert(3) 4 1 + 2Insert(4) 4 1Insert(5) 8 1 + 4Insert(6) 8 1Insert(7) 8 1Insert(8) 8 1Insert(9) 16 1 + 8
123456789
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Aggregate Analysis
n Insert() operations cost
Average cost of operation = (total cost)/(# operations) < 3
Asymptotically, then, a dynamic table costs the same as a fixed-size table Both O(1) per Insert operation
nnnncn
j
jn
ii 3)12(2
lg
01
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Accounting Analysis
Charge each operation $3 amortized cost Use $1 to perform immediate Insert() Store $2
When table doubles $1 reinserts old item, $1 reinserts another old item Point is, we’ve already paid these costs Upshot: constant (amortized) cost per operation
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Accounting Analysis
Suppose must support insert & delete, table should contract as well as expand Table overflows double it (as before) Table < 1/2 full halve it: BAD IDEA (Why?) Better: Table < 1/4 full halve it Charge $3 for Insert (as before) Charge $2 for Delete
Store extra $1 in emptied slot Use later to pay to copy remaining items to new table when
shrinking table
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The End
David Luebke 71 04/21/23
Exercise 1 Feedback
First, my apologies… Harder than I thought Too late to help with midterm
Proof by substitution: T(n) = T(n/2 + n) + n Most people assumed it was O(n lg n)…why?
Resembled proof from class: T(n) = 2T(n/2 + 17) + n The correct intuition: n/2 dominates n term, so it resembles
T(n) = T(n/2) + n, which is O(n) by m.t. Still, if it’s O(n) it’s O(n lg n), right?
David Luebke 72 04/21/23
Exercise 1: Feedback
So, prove by substitution thatT(n) = T(n/2 + n) + n = O(n lg n)
Assume T(n) cn lg n Then T(n) c(n/2 + n) lg (n/2 + n)
c(n/2 + n) lg (n/2 + n) c(n/2 + n) lg (3n/2)
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