m Maximize Σ di = s.t. i=1 WTwo 1+8; for all i with y₁ = 1 Wi+wo-di for all i with y₁ = 0 -Lw;≤LER, 8; ≥ 0 Here, instead of trying to maximize the buffer, we try to find coefficients that map/sepa- rate dat a either above 1 or below 0, with our objective to maximize this separation. We want to make each d; as large and positive as possible so that the real events get a predicted value of 1+8; 1 and false events get a predicted value of -80. L provides an appropriate bound on the coefficients of the hyperplane which you may need to try different values of to experiment. If you're not sure where to start, try L = 5.

m Maximize Σ di = s.t. i=1 WTwo 1+8; for all i with y₁ = 1 Wi+wo-di for all i with y₁ = 0 -Lw;≤LER, 8; ≥ 0 Here, instead of trying to maximize the buffer, we try to find coefficients that map/sepa- rate dat a either above 1 or below 0, with our objective to maximize this separation. We want to make each d; as large and positive as possible so that the real events get a predicted value of 1+8; 1 and false events get a predicted value of -80. L provides an appropriate bound on the coefficients of the hyperplane which you may need to try different values of to experiment. If you're not sure where to start, try L = 5.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Recall that we discussed the "leader-based" deterministic distributed algorithm for 2-colouring a path, that runs in time O(n) on an n node path network. Suppose we now relax the constraint of 2-colouring to mean, it is okay for one of the neighbours of a node to x get the same colour as the mode x, as long as the other neighbour gets a different colour. The end points must get a different colour from their respective unique neighbours. The running time for a best efficiency deterministic distributed algorithm, using unique identifiers for nodes, for this problem would be:
Establish the Shortest-Paths Optimality Conditions in the Proposition P. Then, for any v reachable from s, the value of distTo[v] is the length of some path from s to v, with distTo[v] equal to infinity for all v not reachable from s. Let G be an edge-weighted digraph. s is a source vertex in G. distTo[] is a vertex-indexed array of path lengths in G. These numbers represent the lengths of the shortest pathways if and only if each edge e from v to w satisfies the condition that distTo[w] = distTo[v] + e.weight() (i.e., no edge is eligible).
In the Erdös-Rényi random network model, suppose N=101 and p=1/20, that is, there are 101 vertices, and every pair of vertices has a probability of 1/20 of being connected by an edge. For the network model given what is the probability that a network generated with those parameters has exactly 400 edges? No need to give the decimal value, the mathematical expression will suffice
rate(F, C) = cap(C): To prove this, it is sufficient to prove that everyedge u, v crossing from U to V has flow in F at capacity (Fu,v = cu,v) andevery edge v, u crossing back from V to U has zero flow in F. These give thatrate(F, C) = u∈Uv∈V [Fu,v − Fv,u] = u∈Uv∈V [cu,v − 0] = cap(C).
Prove that (Generic shortest-paths algorithm) Proposition Q Initialise all distTo[] values to infinity and distTo[s] to 0, then carry out the following: Every edge in G should be relaxed until none are left. The value of distTo[w] after this calculation is the length of the shortest path from s to w (and the value of edgeTo[] is the final edge on that path) for all vertices w accessible from s.
Suppose there is undirected graph F with nonnegative edge weights we ≥ 0. You have also calculated the minimum spanning tree of F and also the shortest paths to all nodes from a particular node p ∈ V . Now, suppose that each edge weight is increased by 1, so the new weights are we′ = we + 1. (a) Will there be a change of the minimum spanning tree? Provide an example where it does or prove that it cannot change. (b) (3 points) Will the shortest paths from p change? Provide an example where it does or prove that it cannot change.
Consider the network in the following figure. Edges that are not pictured have a length of ∞. Image attatched (a) What is the optimal TSP tour on this network, and what is its total length, z*? What is the optimal 1-tree Tˆ* on this network (“rooted” at node 1), and what is its total length, z(Tˆ*)? (b) Find the best Held-Karp lower bound you can. That is, find weights to add to one or more nodes so that z* and the right-hand side of the following formula are as close as possible. z* ≥ z(Tˆ*) + SUM{i∈N} λi(di(Tˆ*) − 2) Image attatched Here Tˆ* is the optimal 1-tree under the revised distance matrix, z(Tˆ*) is its cost and di(Tˆ*) is the degree of node i in 1-tree Tˆ*.
Develop an SP client that performs a sensitivity analysis on the edge-weighted digraph’s edges with respect to a given pair of vertices s and t: Compute a V-by-V boolean matrix such that, for every v and w, the entry in row v and column w is true if v->w is an edge in the edge-weighted digraphs whose weight can be increasedwithout the shortest-path length from v to w being increased and is false otherwise.
(Shortest-paths optimality criteria) Demonstrate Proposition P. Let G be an edge-weighted digraph, with s as a source vertex in G and distTo[] a vertex-indexed array of path lengths in G such that the value of distTo[v] is the length of some path from s to v for all v accessible from s, and distTo[v] equal to infinity for all v not reachable from s. These are the lengths of the shortest routes if and only if they meet distTo[w] = distTo[v] + e.weight() for each edge e from v to w (or no edge is eligible).
Prove: Proposition Q. ( Generic shortest-paths algorithm) Initialize distTo[s] to 0 andall other distTo[] values to infinity, and proceed as follows:Relax any edge in G, continuing until no edge is eligible.For all vertices w reachable from s, the value of distTo[w] after this computationis the length of a shortest path from s to w (and the value of edgeTo[] is the lastedge on that path).
Suppose that you want to get from vertex s to vertex t in an unweighted graph G = (V, E), but you would like to stop by vertex u if it is possible to do so without increasing the length of your path by more than a factor of α. Describe an efficient algorithm that would determine an optimal s-t path given your preference for stopping at u along the way if doing so is not prohibitively costly. (It should either return the shortest path from s to t or the shortest path from s to t containing u, depending on the situation)
Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. Consider the following greedy algorithm for generating maximal independent sets: starting with an empty set I, process the vertices in V one at a time, adding v to I is v is not connected to any vertex already in I. 2) Argue that the output I of this algorithm is a maximal independent set.