We study the problem of estimating the value of sums of the form Sp≜∑(xip) when one has the ability to sample xi≥0 with probability proportional to its magnitude. When p=2, this problem is equivalent to estimating the selectivity of a self-join query in database systems when one can sample rows randomly. We also study the special case when {xi} is the degree sequence of a graph, which corresponds to counting the number of p-stars in a graph when one has the ability to sample edges randomly. Our algorithm for a (1±ε)-multiplicative approximation of Sp has query and time complexities O(mloglognϵ2S1/pp). Here, m=∑xi/2 is the number of edges in the graph, or equivalently, half the number of records in the database table. Similarly, n is the number of vertices in the graph and the number of unique values in the database table. We also provide tight lower bounds (up to polylogarithmic factors) in almost all cases, even when {xi} is a degree sequence and one is allowed to use the structure of the graph to try to get a better estimate. We are not aware of any prior lower bounds on the problem of join selectivity estimation. For the graph problem, prior work which assumed the ability to sample only vertices uniformly gave algorithms with matching lower bounds (Gonen et al. in SIAM J Comput 25:1365–1411, 2011). With the ability to sample edges randomly, we show that one can achieve faster algorithms for approximating the number of star subgraphs, bypassing the lower bounds in this prior work. For example, in the regime where Sp≤n, and p=2, our upper bound is O~(n/S1/2p), in contrast to their Ω(n/S1/3p) lower bound when no random edge queries are available. In addition, we consider the problem of counting the number of directed paths of length two when the graph is directed. This problem is equivalent to estimating the selectivity of a join query between two distinct tables. We prove that the general version of this problem cannot be solved in sublinear time. However, when the ratio between in-degree and out-degree is bounded—or equivalently, when the ratio between the number of occurrences of values in the two columns being joined is bounded—we give a sublinear time algorithm via a reduction to the undirected case.
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Sublinear Algorithms and Lower Bounds for Metric TSP Cost Estimation
We consider the problem of designing sublinear time algorithms for estimating the cost of minimum] metric traveling salesman (TSP) tour. Specifically, given access to a n × n distance matrix D that specifies pairwise distances between n points, the goal is to estimate the TSP cost by performing only sublinear (in the size of D) queries. For the closely related problem of estimating the weight of
a metric minimum spanning tree (MST), it is known that for any epsilon > 0, there exists an O^~(n/epsilon^O(1))-time algorithm that returns a (1+epsilon)-approximate estimate of the MST cost. This result immediately
implies an O^~(n/epsilon^O(1)) time algorithm to estimate the TSP cost to within a (2 + epsilon) factor for any
epsilon > 0. However, no o(n^2)-time algorithms are known to approximate metric TSP to a factor that is
strictly better than 2. On the other hand, there were also no known barriers that rule out existence of (1 + epsilon)-approximate estimation algorithms for metric TSP with O^~ (n) time for any fixed epsilon > 0. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost.
On the algorithmic side, we first consider the graphic TSP problem where the metric D corresponds to shortest path distances in a connected unweighted undirected graph. We show that there exists an O^~(n) time algorithm that estimates the cost of graphic TSP to within a factor of (2 − epsilon_0) for some epsilon_0 > 0. This is the first sublinear cost estimation algorithm for graphic TSP that
achieves an approximation factor less than 2. We also consider another well-studied special case of metric TSP, namely, (1, 2)-TSP where all distances are either 1 or 2, and give an O^~(n ^ 1.5) time algorithm to estimate optimal cost to within a factor of 1.625. Our estimation algorithms for graphic
TSP as well as for (1, 2)-TSP naturally lend themselves to O^~(n) space streaming algorithms that give an 11/6-approximation for graphic TSP and a 1.625-approximation for (1, 2)-TSP. These results
motivate the natural question if analogously to metric MST, for any epsilon > 0, (1 + epsilon)-approximate
estimates can be obtained for graphic TSP and (1, 2)-TSP using O^~ (n) queries. We answer this
question in the negative – there exists an epsilon_0 > 0, such that any algorithm that estimates the cost of
graphic TSP ((1, 2)-TSP) to within a (1 + epsilon_0)-factor, necessarily requires
(n^2) queries. This lower bound result highlights a sharp separation between the metric MST and metric TSP problems. Similarly to many classical approximation algorithms for TSP, our sublinear time estimation algorithms utilize subroutines for estimating the size of a maximum matching in the underlying
graph. We show that this is not merely an artifact of our approach,
and that for any epsilon > 0, any
algorithm that estimates the cost of graphic TSP or (1, 2)-TSP to within a (1 + epsilon)-factor, can also
be used to estimate the size of a maximum matching in a bipartite graph to within an epsilon n additive
error. This connection allows us to translate known lower bounds for matching size estimation in
various models to similar lower bounds for metric TSP cost estimation.
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- Award ID(s):
- 1733794
- NSF-PAR ID:
- 10185976
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 168
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 30:0 - 30:18
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ICALP.2020.30
