Question

Problem 1 (40pt, 5pt each). In class we proved the inequality 2m ≤ K(G) ≤ λ(G) ≤ δ(G) ≤ (2m)/(n), where K(G) is the vertex-connectivity of G, λ(G) is the edge-connectivity, δ(G) is the minimum degree, and m and n are respectively the number of edges and vertices of G. In this problem, you will prove that for 'most' graphs G, the four values in the inequality are very close together. In particular, you will prove that for a large randomly generated graph G, it is very likely that all four of the values are between 0.49n and 0.51n. The following theorem will likely be useful. It gives a bound on how likely it is that you get 'about' (K)/2 heads when you flip K fair coins. Theorem ((Special case of) Hoeffding's Inequality-you may use this without proof). Suppose you flip K fair coins, and you get H heads. Then for any constant c, the probability that H>((1)/2+c)K and the probability that H<((1)/2-c)K are at most Problem 1 (40pt, 5pt each). In class we proved the inequality 2m ≤ K(G) ≤ λ(G) ≤ δ(G) ≤ n where K(G) is the vertex-connectivity of G, λ(G) is the edge-connectivity, δ(G) is the minimum degree, and m and n are respectively the number of edges and vertices of G. In this problem, you will prove that for ‘most' graphs G, the four values in the inequality are very close together. In particular, you will prove that for a large randomly generated graph G, it is very likely that all four of the values are between 0.49n and 0.51n. The following theorem will likely be useful. It gives a bound on how likely it is that you get ‘about' K/2 heads when you flip K fair coins. Theorem ((Special case of) Hoeffding's Inequality-you may use this without proof). Suppose you flip K fair coins, and you get H heads. Then for any constant c, the probability that H > (1/2+c)K and the probability that H < (1/2 – c)K are at most e-2eK. For instance, the probability that if you flip K fair coins you get more than .55K heads is at most e -20.05)PK = e-005K. The probability that you get less than .45K heads is also at most e-.005K Let G be a randomly generated graph on n vertices, in the following way: for each of the () possible edges between n vertices, a fair coin is flipped. If it comes up heads, an edge is drawn in that location. If it comes up tails, no edge is drawn. So each individual edge will appear with probability 1/2. (G is called the Erdős-Rényi random graph G(n,1/2), as there are n vertices and each possible edge is chosen with probability 1/2.) The actual parts of the problem begin below. (a) Let m be the number of edges of G. Using Hoeffding's Inequality, prove that the probability that m > 0.255n2 is at most e-nº/10000. (Hint: You're flipping (%) coins, one for each of the edges-but (%) < so the probability that you get at least .255n2 heads when flipping (2) coins is at most the probability that you get at least .255nº heads when flipping coins. Figure out the correct parameters in Hoeffding's inequality to prove this last statement.) (b) Show that there are 3" ways to choose two (possibly empty) disjoint sets of vertices A and B. < (c) Show that if (G) < 0.49N, then there are two disjoint nonempty sets A and B of vertices in G such that A 0.51n, B > n/4, and A and B have no edges between them. (Hint: Let S be a minimal vertex cut and let A be the smallest connected component of G - S.) Call such a pair (A, B) a bad pair. (d) Show that if A and B are disjoint sets of vertices with Al > 8, the probability that (A,B) is a bad pair is at most 4-". (Hint: If |B 1600, prove that there is a vertex of G that is nonadjacent to at least .505n other vertices. (Hint: Prove B > .505n.) (f) Using Hoeffding's Inequality, prove that the probability that a particular vertex of G is nonad- jacent to at least .505n other vertices is at most e-n/20000 Conclude that the probability that at least one vertex of G is nonadjacent to at least .505n other vertices is at most ne-n/20000 Putting everything together: (g) Let n > 1600. Using parts (e) and (f), show that the probability that there exists a bad pair (A, B) with |A| < 8 is at most ne-n/20000. Using parts (b) and (d), show that the probability that the probability that there exists a bad pair (A,B) with Al> 8 is at most (3/4)". Use these two results and part (c) to show that the probability that K(G) < 0.49N is at most (3/4)" + ne-n/20000 (h) Using Parts (a) and (g), show that if n > 1600, the probability that .49n

Name: problem 1 40pt 5pt each in class we proved the inequality 2m kg g g 2mn where kg is the vertex connectivity of g g is the edge connectivity g is the minimum degree and m and n are respective 13496
Uploaded: 2023-02-28T06:21:54-08:00
Duration: 3 min 57 s
Channel: Dominador Tan
Description: problem 1 40pt 5pt each in class we proved the inequality 2m kg g g 2mn where kg is the vertex connectivity of g g is the edge connectivity g is the minimum degree and m and n are respective 13496

          Problem 1 (40pt, 5pt each). In class we proved the inequality
2m ≤ K(G) ≤ λ(G) ≤ δ(G) ≤ (2m)/(n),
where K(G) is the vertex-connectivity of G, λ(G) is the edge-connectivity, δ(G) is the minimum
degree, and m and n are respectively the number of edges and vertices of G.
In this problem, you will prove that for 'most' graphs G, the four values in the inequality are
very close together. In particular, you will prove that for a large randomly generated graph G, it is
very likely that all four of the values are between 0.49n and 0.51n.
The following theorem will likely be useful. It gives a bound on how likely it is that you get
'about' (K)/2 heads when you flip K fair coins.
Theorem ((Special case of) Hoeffding's Inequality-you may use this without proof). Suppose you
flip K fair coins, and you get H heads. Then for any constant c, the probability that H>((1)/2+c)K
and the probability that H<((1)/2-c)K are at most Problem 1 (40pt, 5pt each). In class we proved the inequality 2m ≤ K(G) ≤ λ(G) ≤ δ(G) ≤ n where K(G) is the vertex-connectivity of G, λ(G) is the edge-connectivity, δ(G) is the minimum degree, and m and n are respectively the number of edges and vertices of G. In this problem, you will prove that for ‘most' graphs G, the four values in the inequality are very close together. In particular, you will prove that for a large randomly generated graph G, it is very likely that all four of the values are between 0.49n and 0.51n. The following theorem will likely be useful. It gives a bound on how likely it is that you get ‘about' K/2 heads when you flip K fair coins. Theorem ((Special case of) Hoeffding's Inequality-you may use this without proof). Suppose you flip K fair coins, and you get H heads. Then for any constant c, the probability that H > (1/2+c)K and the probability that H < (1/2 – c)K are at most e-2eK. For instance, the probability that if you flip K fair coins you get more than .55K heads is at most e -20.05)PK = e-005K. The probability that you get less than .45K heads is also at most e-.005K Let G be a randomly generated graph on n vertices, in the following way: for each of the () possible edges between n vertices, a fair coin is flipped. If it comes up heads, an edge is drawn in that location. If it comes up tails, no edge is drawn. So each individual edge will appear with probability 1/2. (G is called the Erdős-Rényi random graph G(n,1/2), as there are n vertices and each possible edge is chosen with probability 1/2.) The actual parts of the problem begin below. (a) Let m be the number of edges of G. Using Hoeffding's Inequality, prove that the probability that m > 0.255n2 is at most e-nº/10000. (Hint: You're flipping (%) coins, one for each of the edges-but (%) < so the probability that you get at least .255n2 heads when flipping (2) coins is at most the probability that you get at least .255nº heads when flipping coins. Figure out the correct parameters in Hoeffding's inequality to prove this last statement.) (b) Show that there are 3" ways to choose two (possibly empty) disjoint sets of vertices A and B. < (c) Show that if (G) < 0.49N, then there are two disjoint nonempty sets A and B of vertices in G such that A 0.51n, B > n/4, and A and B have no edges between them. (Hint: Let S be a minimal vertex cut and let A be the smallest connected component of G - S.) Call such a pair (A, B) a bad pair. (d) Show that if A and B are disjoint sets of vertices with Al > 8, the probability that (A,B) is a bad pair is at most 4-". (Hint: If |B 1600, prove that there is a vertex of G that is nonadjacent to at least .505n other vertices. (Hint: Prove B > .505n.) (f) Using Hoeffding's Inequality, prove that the probability that a particular vertex of G is nonad- jacent to at least .505n other vertices is at most e-n/20000 Conclude that the probability that at least one vertex of G is nonadjacent to at least .505n other vertices is at most ne-n/20000 Putting everything together: (g) Let n > 1600. Using parts (e) and (f), show that the probability that there exists a bad pair (A, B) with |A| < 8 is at most ne-n/20000. Using parts (b) and (d), show that the probability that the probability that there exists a bad pair (A,B) with Al> 8 is at most (3/4)". Use these two results and part (c) to show that the probability that K(G) < 0.49N is at most (3/4)" + ne-n/20000 (h) Using Parts (a) and (g), show that if n > 1600, the probability that .49n

problem 1 40pt 5pt each in class we proved the inequality 2m kg g g 2mn where kg is the vertex connectivity of g g is the edge connectivity g is the minimum degree and m and n are respective 13496

Added by Denise C.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Dominador Tan

02/28/2023

Step 1

255n^2 is at most e^(-n/10000). Show more…

Show all steps

Thanks for your feedback!

Problem 1 (40pt, 5pt each). In class we proved the inequality 2m ≤ K(G) ≤ λ(G) ≤ δ(G) ≤ (2m)/(n), where K(G) is the vertex-connectivity of G, λ(G) is the edge-connectivity, δ(G) is the minimum degree, and m and n are respectively the number of edges and vertices of G. In this problem, you will prove that for 'most' graphs G, the four values in the inequality are very close together. In particular, you will prove that for a large randomly generated graph G, it is very likely that all four of the values are between 0.49n and 0.51n. The following theorem will likely be useful. It gives a bound on how likely it is that you get 'about' (K)/2 heads when you flip K fair coins. Theorem ((Special case of) Hoeffding's Inequality-you may use this without proof). Suppose you flip K fair coins, and you get H heads. Then for any constant c, the probability that H>((1)/2+c)K and the probability that H<((1)/2-c)K are at most Problem 1 (40pt, 5pt each). In class we proved the inequality 2m ≤ K(G) ≤ λ(G) ≤ δ(G) ≤ n where K(G) is the vertex-connectivity of G, λ(G) is the edge-connectivity, δ(G) is the minimum degree, and m and n are respectively the number of edges and vertices of G. In this problem, you will prove that for ‘most' graphs G, the four values in the inequality are very close together. In particular, you will prove that for a large randomly generated graph G, it is very likely that all four of the values are between 0.49n and 0.51n. The following theorem will likely be useful. It gives a bound on how likely it is that you get ‘about' K/2 heads when you flip K fair coins. Theorem ((Special case of) Hoeffding's Inequality-you may use this without proof). Suppose you flip K fair coins, and you get H heads. Then for any constant c, the probability that H > (1/2+c)K and the probability that H < (1/2 – c)K are at most e-2eK. For instance, the probability that if you flip K fair coins you get more than .55K heads is at most e -20.05)PK = e-005K. The probability that you get less than .45K heads is also at most e-.005K Let G be a randomly generated graph on n vertices, in the following way: for each of the () possible edges between n vertices, a fair coin is flipped. If it comes up heads, an edge is drawn in that location. If it comes up tails, no edge is drawn. So each individual edge will appear with probability 1/2. (G is called the Erdős-Rényi random graph G(n,1/2), as there are n vertices and each possible edge is chosen with probability 1/2.) The actual parts of the problem begin below. (a) Let m be the number of edges of G. Using Hoeffding's Inequality, prove that the probability that m > 0.255n2 is at most e-nº/10000. (Hint: You're flipping (%) coins, one for each of the edges-but (%) < so the probability that you get at least .255n2 heads when flipping (2) coins is at most the probability that you get at least .255nº heads when flipping coins. Figure out the correct parameters in Hoeffding's inequality to prove this last statement.) (b) Show that there are 3" ways to choose two (possibly empty) disjoint sets of vertices A and B. < (c) Show that if (G) < 0.49N, then there are two disjoint nonempty sets A and B of vertices in G such that A 0.51n, B > n/4, and A and B have no edges between them. (Hint: Let S be a minimal vertex cut and let A be the smallest connected component of G - S.) Call such a pair (A, B) a bad pair. (d) Show that if A and B are disjoint sets of vertices with Al > 8, the probability that (A,B) is a bad pair is at most 4-". (Hint: If |B 1600, prove that there is a vertex of G that is nonadjacent to at least .505n other vertices. (Hint: Prove B > .505n.) (f) Using Hoeffding's Inequality, prove that the probability that a particular vertex of G is nonad- jacent to at least .505n other vertices is at most e-n/20000 Conclude that the probability that at least one vertex of G is nonadjacent to at least .505n other vertices is at most ne-n/20000 Putting everything together: (g) Let n > 1600. Using parts (e) and (f), show that the probability that there exists a bad pair (A, B) with |A| < 8 is at most ne-n/20000. Using parts (b) and (d), show that the probability that the probability that there exists a bad pair (A,B) with Al> 8 is at most (3/4)". Use these two results and part (c) to show that the probability that K(G) < 0.49N is at most (3/4)" + ne-n/20000 (h) Using Parts (a) and (g), show that if n > 1600, the probability that .49n