Algorithms for the Generalized Sorting Problem

Q: How can the authors prove that the lowest rank vertices are active?

Note that the lowest rank vertices can have in-edges only from other lowest rank vertices at rate p.Lemma 4.2: For each subset of vertices U ⊆ V such that |U | ≥ 4 log n/p, with probability at least 1 − O( 1n3 ), the authors have that for every vertex v ∈ V , E(v, U), the number of edges between v and the vertices in U is at least 1 and at most 2p |U |.

Question

1. What contributions have the authors mentioned in the paper "Algorithms for the generalized sorting problem" ?

2. How can the authors estimate the in-degree of every vertex?

3. How many probes are needed to improve the lower bound?

4. What is the simplest way to reduce the number of linear extensions?

Accepted Answer

The authors study the generalized sorting problem where they are given a set of n elements to be sorted but only a subset of all possible pairwise element comparisons is allowed.. The generalized sorting problem may be equivalently viewed as follows.. An extensively studied special case of the generalized sorting problem is the nuts and bolts problem where the allowed comparison graph is a complete bipartite graph between two equal-size sets.. The authors also study the sorting problem in randomly generated allowed comparison graphs, and show that when the edge probability is p, Õ ( min { n/p2, n3⁄2p √p } ) comparisons suffice on average to sort.

Accepted Answer

By using the subroutine INIT-SAMPLE with sampling rate 4/ √ np, the authors can estimate the in-degree of every vertex up to an additive error of √ np log n using O(n3/2 √ p log n) probes.

Accepted Answer

The fact that random graphs may be easier than arbitrary graphs suggests that careful combinatorial constructions may be needed to improve the lower bound beyond Ω(n log n) probes.

Accepted Answer

and probing a balanced edge reduces the number of linear extensions by at least a ( 1− 1e √ n) factor, the authors only need toprobe O(n3/2 log n) edges to reduce the number of linear extensions to one, and hence sort the n elements.

Accepted Answer

An unprobed edge is balanced if probing the edge would reduce the number of linear extensions by at least a ( 1− 1e √ n) factor, regardless of theunderlying direction of the edge.

Accepted Answer

For an arbitrary graph of comparisons with the promiseof a directed Hamilton path, the authors can sort with Õ(n3/2) probes with high probability.

Accepted Answer

If the directed graph is arbitrary, then arguably the best the authors can do is uniform sampling which requires sampling at rate Ω(1), and hence uses Ω(n2) probes.

Accepted Answer

The structured cost models assume that the cost of a comparison is a monotone function of the known ‘size’ of the elements being compared.

Accepted Answer

The input to the generalized sorting problem is an undirected graph G(V,E) where an edge (u, v) ∈ E indicates that the element u can be compared to the element v. By probing (u, v), the authors reveal a directed edge (u, v) or (v, u), depending on whether u < v or v < u. Let ~G(V, ~E) denote the directed graph where (u, v) ∈ ~E iff (u, v) ∈ E, and u < v. Given the promise that ~G(V, ~E) is an acyclic graph that contains a directed Hamiltonian path, the problem is to find this path by adaptively probing the smallest number of edges in E.

Accepted Answer

Note that the lowest rank vertices can have in-edges only from other lowest rank vertices at rate p.Lemma 4.2: For each subset of vertices U ⊆ V such that |U | ≥ 4 log n/p, with probability at least 1 − O( 1n3 ), the authors have that for every vertex v ∈ V , E(v, U), the number of edges between v and the vertices in U is at least 1 and at most 2p |U |.

Accepted Answer

By union bound, with probability at least 1− 1 n there are no bad iterations among the first O(n log n) iterations of the while loop.

Accepted Answer

For any three vertices u, v, and w, the authors say w is sandwiched between u and v if (u,w), (w, v) ∈ ~E.Lemma 4.1: With high probability, for every pair of vertices u and v such that π(v)−π(u) ≥ 64 log n/p3, the authors have the following properties: (a) There are at least p2(π(v) − π(u))/2 vertices and atmost 2p2(π(v)−π(u)) vertices sandwiched between u and v. Moreover, at least one quarter of the vertices sandwiched between u and v have rank in the range [ 34π(u) + 1 4π(v), 1 4π(u) + 3 4π(v)].

Accepted Answer

The authors give the first non-trivial algorithms for the generalized sorting problem where only a subset of all pairwise comparisons are allowed.

Accepted Answer

Lemma 3.1: With probability at least 1− 1n2 , (n + 1)x̄v in the procedure in Figure 1 approximates the average rank of each vertex v up to an additive error of 1 for all v ∈ V .

Accepted Answer

The authors will also verify the relation u < v− for every vertex u with rank π(u) < π(v−)−64 log n/p3 using the same recursive procedure the authors used to do this for v. Similarly the authors will do this for v+.

Accepted Answer

Unlike the nuts and bolts problem, since the graph here is adversarially chosen, it may not have large expander subgraphs, and even if it does, it is not clear whether the comparisons in this subgraph are useful in making progress, since an adversary could potentially ensure that any bipartite expanders have vertices of widely different ranks on the left and right sides.

Accepted Answer

By Lemma 4.1, the rank difference between v and the lower marker v− decreases by at least a 34 factor with probability at least 1 4 .

Accepted Answer

Using a recursion tree approach to analyze the recurrence relation for probe complexity of this part and noting that the number of leaves in the tree is O(np3/ log n) the authors find that the probe complexity of all partition steps is O(n log3 n+ n log n/p2).

Accepted Answer

Lemma 3.4: The total number of probes in the binary search algorithm in Figure 3 is at most O(n log n), and the potential ΦV decreases by at least k ≥ √ n.

Algorithms for the Generalized Sorting Problem

Chat with Paper

AI Agents for this Paper

Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Algorithms for the generalized sorting problem" ?

2. How can the authors estimate the in-degree of every vertex?

3. How many probes are needed to improve the lower bound?

4. What is the simplest way to reduce the number of linear extensions?

5. What is the simplest way to determine if an edge is balanced?

6. How many probes can be used to sort a graph of comparisons?

7. What is the way to sample a graph?

8. What is the main idea of structured cost models?

9. How many edges can be found in the generalized sorting problem?

10. How can the authors prove that the lowest rank vertices are active?

11. How many bad iterations of the while loop are there?

12. how many vertices are sandwiched between u and v?

13. What is the first non-trivial algorithm for the generalized sorting problem?

14. What is the procedure for calculating the average rank of each vertex?

15. How can the authors verify the relation u v for every vertex u?

16. What is the problem with the graph of allowed comparisons?

17. How many nps does the rank difference between v and the lower marker v?

18. How do the authors find the probe complexity of all partition steps?

19. how many probes are there in the binary search algorithm?

Figures

Citations

Skyline Queries with Noisy Comparisons

A comparative Study of Sorting Algorithms Comb, Cocktail and Counting Sorting

Vertex ordering problems in directed graph streams

Shattering, Graph Orientations, and Connectivity

On Generalized Comparison-Based Sorting Problems

References

A random polynomial-time algorithm for approximating the volume of convex bodies

An 0(n log n) sorting network

A random polynomial time algorithm for approximating the volume of convex bodies

How good is the information theory bound in sorting

Query strategies for priced information

Related Papers (5)

Perfect Matchings in \~O(n^{1.5}) Time in Regular Bipartite Graphs

Counting H-colorings of partial k-trees

An Approximation Algorithm for the Maximum Independent Set Problem on Planar Graphs

The max quasi-independent set problem

Coloring sparse random k-colorable graphs in polynomial expected time