Information-theoretic Bound Research Articles

Single quantum querying of a database

We present a class of fast quantum algorithms, based on Bernstein and Vazirani's parity problem, that retrieves the entire contents of a quantum database $Y$ in a single query. The class includes binary search problems and coin-weighing problems. We compare the efficiency of these quantum algorithms with the classical algorithms that are bounded by the classical information-theoretic bound. We show the connection between classical algorithms based on several compression codes and our quantum-mechanical method.

Quantum Computers Can Search Arbitrarily Large Databases by a Single Query

This paper shows that a quantum mechanical algorithm that can query information relating to multiple items of the database, can search a database in a single query (a query is defined as any question to the database to which the database has to return a (YES/NO) answer). A classical algorithm will be limited to the information theoretic bound of at least O(log N) queries (which it would achieve by using a binary search).

Edge search in hypergraphs

Consider an undirected hypergraph H = ( X, E) with a probability distribution P on the set E of its hyperedges. We investigate the average case complexity L( H, P) of finding an unknown hyperedge e ∗ ϵ E , chosen according to P, if only tests are allowed that check, whether e ∗ is contained in the induced subhypergraphs H[ Y] for Y ⊂ X, or not. In this note we prove that for hypergraphs with average size of the hyperedges r H( P) ⪯ L( H, P) < H( P) + 3 r. Here H(·) denotes the usual entropy function. The lower bound is the information theoretic bound, whereas the upper bound stems from a construction presented in this note. More precisely, our construction gives search lengths which are bounded from above by − log 2 P(e ∗) + 3 · ¦e ∗¦ for all possible choices of e ∗ .

Search problems for two irregular coins with incomplete feedback: the underweight model

Problems of detecting two (or one of two) irregular coins x and y among a set of n coins are considered. The testing device is such that it returns feedback 1 if x belongs to the test set and y does not, while it returns feedback 0 otherwise. We present a lower bound on the worst-case number of tests necessary to find one of the two irregular coins as well as on the worst-case number of tests to find both of them. The lower bound improves on the information theoretic bound and shows that a “natural” algorithm to find one coin is optimal for infinitely many values of n.

A ternary search problem on graphs

In a previous paper, Aigner studied the following search problem on graphs. For a graph G, let e ∗ϵE(G) be an unknown edge. In order to find e ∗ , we choose a sequence of test-sets A⊆ V( G) where after every test we are told whether e ∗ has both end-vertices in A, one end-vertex, or none. Find the minimum c( G) of tests required. Since in this problem ternary tests are performed, we have the usual information theoretic bound ⌜log 3|E(G)⌝≤c(G) . Beside his main results which are on complete and complete bipartite graphs, Aigner proved that each forest F with maximum degree at most two is optimal, i.e., the information theoretic bound is achieved. In the present paper we consider the more general question, how close we can come to achieving the information theoretic bound for forests with maximum degree at most r, r=1,2,…. Let F r be the class of forests with non-empty edge-set and maximum degree at most r. We shall investigate the function f(r)=max{c(F)-⌜log 3|E(F)|⌝:Fϵ F r} and obtain the result that f(r)=t+1-⌜log 3(2 t+1)⌝ for 2 t < r≤2 t+1 , t=0,1,…. In addition, we show that, with the exception of five small graphs, all members of F 3 are optimal, and we conjecture that a similar result holds for F r, r≥4 .

Detecting the two complementary defectives with a balance scale

We consider the problem of finding a pair of irregular boxes from a set of n boxes using a balance scale. One irregular box is heavier and the other lighter than a regular box but the total weight of the two irregular boxes is the same as the total weight of two regular boxes. Let N(w) denote the maximum number of boxes w weighings can handle. We give a weighing scheme such that N(2t) ≥3t for t ≥2 and N(2t + 1) ≥ 5.3t-1 for t ≥ 1. The N(2t) result meets the information-theoretic bound and hence is the best possible.

The Information-Theoretic Bound is Good for Merging

Let $A = (a_1 > \cdots > a_m )$ and $B = (b_1 > \cdots > b_n )$ be given ordered lists: also let there be given some order relations between $a_i $’s and $b_j $’s. Suppose that an unknown total order exists on $A \cup B$ which is consistent with all these relations ($ = a$ linear extension of the partial order) and we wish to find out this total order by comparing pairs of elements $a_t :b_s $. If the partial order has N linear extensions, then the Information Theoretic Bound says that $\log _2 N$ steps will be required in the worst case from any such algorithm. In this paper we show that there exists an algorithm which will take no more than $C\log _2 N$ comparisons where $C = (\log _2 ((\sqrt 5 + 1)/ 2))^{ -1} $. The computation required to determine the pair $a_t :b_s $ to be compared has length polynomial in $(m + n)$. The constant C is best possible. Many related results are reviewed.

Compact storage schemes for formatted files by spanning trees

The use of spanning trees in the compression of data files is studied. A new upper bound for the length of the minimal spanning tree, giving the size of the compressed file, is derived. A special front compression technique is proposed for unordered files. The space demands are compared to an information theoretical lower bound of the file size.

A Bound on Processor Gain

In data transmission systems using electromagnetic propagation there is often interest in the antijam capability of the system and the probability that the transmitted signal can be intercepted. It is shown that, neglecting antenna gain, both of these are adequately described by the processor gain. An information theoretic bound is given for the processor gain and the performance of several conventional signaling systems is indicated. Improvement using forward error correction is also investigated.

Capacity of Classes of Gaussian Channels

Gaussian channels capacity, studying supremum of information transmission rates with small error probability