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233-234233-234 Sedgewick & Wayne (2004); Chazelle (2005) Sedgewick & Wayne (2004); Chazelle (2005) chazelle@cs.princeton.educhazelle@cs.princeton.edu
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computation theory experimentation
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computation theory experimentation
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computation theory experimentation
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computation theory experimentation Moore’s Law
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No Moore’s Law there ! there ! No Moore’s Law there ! there ! Inverse Moore’s Law?
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1. Power of algorithms
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2. Duality: data & programs 3. Recursion 1. Power of algorithms
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A program in C /**/char q='"',*a="*//**/char q='%c',*a=%c%s%c*/};)b(stup;]d[b=]d-472 [b)--d(elihw;)q,a,q,q,2+a,b(ftnirps;)b(stup{)(niam;731=d tni;]572 [b,",b[275];int d=137;main(){puts(b);sprintf(b,a+2,q,q,a,q);while(d--) b[274-d]=b[d];puts(b);}/*c%s%c%=a*,'c%'=q rahc/**//*"=a*,'"'=q rahc/**/ (By Dan Hoey)
![A program in C /**/char q= ,*a= *//**/char q= %c ,*a=%c%s%c*/};)b(stup;]d[b=]d-472 [b)--d(elihw;)q,a,q,q,2+a,b(ftnirps;)b(stup{)(niam;731=d tni;]572 [b, ,b[275];int d=137;main(){puts(b);sprintf(b,a+2,q,q,a,q);while(d--) b[274-d]=b[d];puts(b);}/*c%s%c%=a*, c% =q rahc/**//* =a*, =q rahc/**/ (By Dan Hoey)](http://images.slideplayer.com./16/4997721/slides/slide_10.jpg "A program in C /**/char q= ,*a= *//**/char q= %c ,*a=%c%s%c*/};)b(stup;]d[b=]d-472 [b)--d(elihw;)q,a,q,q,2+a,b(ftnirps;)b(stup{)(niam;731=d tni;]572 [b, ,b[275];int d=137;main(){puts(b);sprintf(b,a+2,q,q,a,q);while(d--) b[274-d]=b[d];puts(b);}/*c%s%c%=a*, c% =q rahc/**//* =a*, =q rahc/**/ (By Dan Hoey)")
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char*s="char*s=%c%s%c;void main(){ printf(s,34,s,34);}";void main(){printf(s,34,s,34);} A program in C
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'; printf ($s,39,$s,39); ?> A program in php
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$s="$s=%c%s%c;printf$s,34,$s,34;";printf$s,34,$s,34; A program in perl
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l='l=%s;print l%`l`';print l%`l` A program in python
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class S{public static void main(String[]a){String s="class S{ public static void main(String[]a){String s=;char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}}"; char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}} A program in Java Output = ? class S{public static void main(String[]a){String s="class S{ public static void main(String[]a){String s=;char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}}"; char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}}
![class S{public static void main(String[]a){String s= class S{ public static void main(String[]a){String s=;char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}} ; char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}} A program in Java Output = .](http://images.slideplayer.com./16/4997721/slides/slide_15.jpg "class S{public static void main(String[]a){String s= class S{ public static void main(String[]a){String s=;char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}} ; char c=34; System.out.println(s.substring(0,52)+c+s+c+s.substring(52));}}.")
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Why does Schroedinger hate cats? Print next statement twice Output = ? Why does Schroedinger hate cats? A B AB BB
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Print next statement twice Output = ? Set B=A AA AA Print next statement twice
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string of symbols datacommand
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Rogozhin (1996) : 24 states input data output data program data Babbage (1820)
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Print next statement twice Why does Schroedinger hate cats? Print next statement twice
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duality recursion
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duality + recursion Self-reproduction =
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http://www.accessexcellence.org/RC/VL/GG/dna_replicating.html duality recursion gene - protein dble-stranded
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What is the difference between classical math and computer science ?
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Jos Leys
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Protein Interaction Network Jeong et al
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Adjacency matrix
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Adjacency lists
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public class Graph { private int V; private Node[] adj; private int[] degree; private static class Node { int vertex; Node next; Node(int v, Node next) { this.vertex = v; this.next = next; } Adjacency List Implementation vertex next Node
![public class Graph { private int V; private Node[] adj; private int[] degree; private static class Node { int vertex; Node next; Node(int v, Node next) { this.vertex = v; this.next = next; } Adjacency List Implementation vertex next Node](http://images.slideplayer.com./16/4997721/slides/slide_31.jpg "public class Graph { private int V; private Node[] adj; private int[] degree; private static class Node { int vertex; Node next; Node(int v, Node next) { this.vertex = v; this.next = next; } Adjacency List Implementation vertex next Node")
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public class Graph { private int V; private Node[] adj; private int[] degree; private static class Node { int vertex; Node next; Node(int v, Node next) { this.vertex = v; this.next = next; } public Graph(int V) { this.V = V; adj = new Node[V]; degree = new int[V]; } Adjacency List Implementation graph on V vertices with no edges Node
![public class Graph { private int V; private Node[] adj; private int[] degree; private static class Node { int vertex; Node next; Node(int v, Node next) { this.vertex = v; this.next = next; } public Graph(int V) { this.V = V; adj = new Node[V]; degree = new int[V]; } Adjacency List Implementation graph on V vertices with no edges Node](http://images.slideplayer.com./16/4997721/slides/slide_32.jpg "public class Graph { private int V; private Node[] adj; private int[] degree; private static class Node { int vertex; Node next; Node(int v, Node next) { this.vertex = v; this.next = next; } public Graph(int V) { this.V = V; adj = new Node[V]; degree = new int[V]; } Adjacency List Implementation graph on V vertices with no edges Node")
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public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v's adjacency list add v to w's adjacency list v w x z y
![public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y](http://images.slideplayer.com./16/4997721/slides/slide_33.jpg "public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y")
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public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v's adjacency list add v to w's adjacency list v w x z y
![public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y](http://images.slideplayer.com./16/4997721/slides/slide_34.jpg "public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y")
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public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v's adjacency list add v to w's adjacency list v w x z y
![public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y](http://images.slideplayer.com./16/4997721/slides/slide_35.jpg "public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y")
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public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v's adjacency list add v to w's adjacency list v w x z y w
![public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y w](http://images.slideplayer.com./16/4997721/slides/slide_36.jpg "public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y w")
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public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v's adjacency list add v to w's adjacency list v w x z y w
![public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y w](http://images.slideplayer.com./16/4997721/slides/slide_37.jpg "public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y w")
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public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v's adjacency list add v to w's adjacency list v w x z y w
![public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y w](http://images.slideplayer.com./16/4997721/slides/slide_38.jpg "public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w x z y w")
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public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v's adjacency list add v to w's adjacency list v w z y w x
![public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w z y w x](http://images.slideplayer.com./16/4997721/slides/slide_39.jpg "public int V() { return V; } public void addEdge(int v, int w) { adj[v] = new Node(w, adj[v]); adj[w] = new Node(v, adj[w]); degree[v]++; degree[w]++; } Adjacency List Implementation # vertices add w to v s adjacency list add v to w s adjacency list v w z y w x")
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public int[] neighbors(int v) { int[] neighbors = new int[degree[v]]; int i = 0; for (Node x = adj[v]; x != null; x = x.next) neighbors[i++] = x.vertex; return neighbors; } Return list of neighbors of v as an array d c a b v neighbors adj null a b c d i= 0i= 1 i= 2i= 3 xx
![public int[] neighbors(int v) { int[] neighbors = new int[degree[v]]; int i = 0; for (Node x = adj[v]; x != null; x = x.next) neighbors[i++] = x.vertex; return neighbors; } Return list of neighbors of v as an array d c a b v neighbors adj null a b c d i= 0i= 1 i= 2i= 3 xx](http://images.slideplayer.com./16/4997721/slides/slide_40.jpg "public int[] neighbors(int v) { int[] neighbors = new int[degree[v]]; int i = 0; for (Node x = adj[v]; x != null; x = x.next) neighbors[i++] = x.vertex; return neighbors; } Return list of neighbors of v as an array d c a b v neighbors adj null a b c d i= 0i= 1 i= 2i= 3 xx")
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1. Birds eat the bread crumbs 2. They don’t random walk DFS/BFS Hansel & Gretel
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Diffusion equation
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Normal distribution Random walk
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With bread crumbs one can find exit in time proportional to V+E DFS/BFS Hansel & Gretel
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Undirected Depth First Search Adjacency Lists A: F C B G B: A C: A D: F E E: G F D F: A E D: G: E A: H: I: I: H: F A BCG DE HI
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack F newly discovered
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) A already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) E newly discovered E
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) G newly discovered G
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) visit(G) (G, E) (G, A) E already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) visit(G) (G, E) (G, A) A already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) visit(G) (G, E) (G, A) Finished G
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) F already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) D newly discovered D
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) visit(D) (D, F) (D, E) F already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) visit(D) (D, F) (D, E) E already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) visit(D) (D, F) (D, E) Finished D
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) visit(E) (E, G) (E, F) (E, D) Finished E
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) D already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(F) (F, A) (F, E) (F, D) Finished F
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack C newly discovered C
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(C) (C, A) A already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(C) (C, A) Finished C
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack B newly discovered B
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(B) (B, A) A already marked
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack visit(B) (B, A) Finished B
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack G already finished
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visit(A) (A, F) (A, C) (A, B) (A, G) Undirected Depth First Search F A BCG DE HI Stack Finished A
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Undirected Depth First Search F A BCG DE HI Stack
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Undirected Depth First Search F A BCG DE HI Stack visit(H) (H, I) I newly discovered I
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Undirected Depth First Search F A BCG DE HI Stack visit(H) (H, I) visit(I) (I, H) H already marked
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Undirected Depth First Search F A BCG DE HI Stack visit(H) (H, I) visit(I) (I, H) Finished I
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Undirected Depth First Search F A BCG DE HI Stack visit(H) (H, I) FinishedH
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Undirected Depth First Search F A BCG DE HI Stack
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public class DFSearcher { private static int UNMARKED = -1; private int[] cc; private int components; public DFSearcher(Graph G) { // next slide } public int component(int v) { return cc[v]; } public int components() { return components; } Connected components of a graph G. * For each vertex, give id number corresponding to component.
![public class DFSearcher { private static int UNMARKED = -1; private int[] cc; private int components; public DFSearcher(Graph G) { // next slide } public int component(int v) { return cc[v]; } public int components() { return components; } Connected components of a graph G.](http://images.slideplayer.com./16/4997721/slides/slide_79.jpg "* For each vertex, give id number corresponding to component..")
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public DFSearcher(Graph G) { components = 0; cc = new int[G.V()]; for (int v = 0; v < G.V(); v++) cc[v] = UNMARKED; for (int v = 0; v < G.V(); v++) { if (cc[v] == UNMARKED) { dfs(G, v); components++; } private void dfs(Graph G, int v) { cc[v] = components; int[] neighbors = G.neighbors(v); for (int i = 0; i < neighbors.length; i++) { int w = neighbors[i]; if (cc[w] == UNMARKED) dfs(G, w); } v c b d a b a c d w 0
![public DFSearcher(Graph G) { components = 0; cc = new int[G.V()]; for (int v = 0; v < G.V(); v++) cc[v] = UNMARKED; for (int v = 0; v < G.V(); v++) { if (cc[v] == UNMARKED) { dfs(G, v); components++; } private void dfs(Graph G, int v) { cc[v] = components; int[] neighbors = G.neighbors(v); for (int i = 0; i < neighbors.length; i++) { int w = neighbors[i]; if (cc[w] == UNMARKED) dfs(G, w); } v c b d a b a c d w 0](http://images.slideplayer.com./16/4997721/slides/slide_80.jpg "public DFSearcher(Graph G) { components = 0; cc = new int[G.V()]; for (int v = 0; v < G.V(); v++) cc[v] = UNMARKED; for (int v = 0; v < G.V(); v++) { if (cc[v] == UNMARKED) { dfs(G, v); components++; } private void dfs(Graph G, int v) { cc[v] = components; int[] neighbors = G.neighbors(v); for (int i = 0; i < neighbors.length; i++) { int w = neighbors[i]; if (cc[w] == UNMARKED) dfs(G, w); } v c b d a b a c d w 0")
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Is graph connected?
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Depth First Search
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