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1974

Volume 3, Issue 4, pp. 255-326


The Number of $1$’s in Binary Integers: Bounds and Extremal Properties

M. D. McIlroy

SIAM J. Comput. 3, pp. 255-261 (7 pages) | Cited 6 times

Online Publication Date: July 13, 2006

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Closed formulas provide tight bounds for $G(n)$, the total number of 1’s in the binary representations of integers less than $n$. This function satisfies an extremal recurrence, which gives the maximum cost of a process that creates a set of $n$ objects by repeatedly merging pairs of smaller sets, starting from $n$ singletons, incurring a cost equal to the size of the smaller set at each merger:\[ G(n) = \max\limits_{1 \leqq i \leqq n /2} [i + G(i) + G(n - i)], \] where $G(1) = 0$. The set of pairs $(i,n - i)$ at which the maximum is attained has an interesting structure.

Computationally Related Problems

Sartaj Sahni

SIAM J. Comput. 3, pp. 262-279 (18 pages) | Cited 53 times

Online Publication Date: July 13, 2006

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We look at several problems from areas such as network flows, game theory, artificial intelligence, graph theory, integer programming and nonlinear programming and show that they are related in that any one of these problems is solvable in polynomial time if all the others are, too. At present, no polynomial time algorithm for these problems is known. These problems extend the equivalence class of problems known as P-Complete. The problem of deciding whether the class of languages accepted by polynomial time nondeterministic Turing machines is the same as that accepted by polynomial time deterministic Turing machines is related to P-Complete problems in that these two classes of languages are the same if each P-Complete problem has a polynomial deterministic solution. In view of this, it appears very likely that this equivalence class defines a class of problems that cannot be solved in deterministic polynomial time.

A Note on Perfect Elimination Digraphs

D. J. Kleitman

SIAM J. Comput. 3, pp. 280-282 (3 pages) | Cited 3 times

Online Publication Date: July 13, 2006

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The purpose of this note is to settle a conjecture on perfect elimination digraphs raised by Haskins and Rose in this journal, and to make some remarks that illuminate the nature of digraphs satisfying the conditions defined by them.

Reversal-Bounded Acceptors and Intersections of Linear Languages

Ronald Book, Maurice Nivat, and Michael Paterson

SIAM J. Comput. 3, pp. 283-295 (13 pages) | Cited 8 times

Online Publication Date: July 13, 2006

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A Turing machine whose behavior is restricted so that each read-write head can change its direction only a bounded number of times is reversal-bounded. Here we consider nondeterministic multitape acceptors which are both reversal-bounded and also operate in linear time. Our main result shows that such an acceptor need have only three pushdown stores as auxiliary storage, each pushdown store need make only one reversal, and the acceptor can operate in real time.

Existence of Graphs with Three Spanning Trees and Given Degree Sequence

Sukhamay Kundu

SIAM J. Comput. 3, pp. 296-298 (3 pages) | Cited 1 time

Online Publication Date: July 13, 2006

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A simple necessary and sufficient condition is given for a degree sequence to be realizable by a graph that contains three mutually edge-disjoint spanning trees.

Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms

D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, and R. L. Graham

SIAM J. Comput. 3, pp. 299-325 (27 pages) | Cited 109 times

Online Publication Date: July 13, 2006

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The following abstract problem models several practical problems in computer science and operations research: given a list $L$ of real numbers between 0 and l, place the elements of $L$ into a minimum number $L^ * $ of “bins” so that no bin contains numbers whose sum exceeds l. Motivated by the likelihood that an excessive amount of computation will be required by any algorithm which actually determines an optimal placement, we examine the performance of a number of simple algorithms which obtain “good” placements. The first-fit algorithm places each number, in succession, into the first bin in which it fits. The best-fit algorithm places each number, in succession, into the most nearly full bin in which it fits. We show that neither the first-fit nor the best-fit algorithm will ever use more than $\frac{17}{10}L^ * + 2$ bins. Furthermore, we outline a proof that, if $L$ is in decreasing order, then neither algorithm will use more than $\frac{11}{9} L^ * + 4$ bins. Examples are given to show that both upper bounds are essentially the best possible. Similar results are obtained when the list $L$ contains no numbers larger than $\alpha < 1$.

Erratum: Genetic Algorithms and the Optimal Allocation of Trials

John H. Holland

SIAM J. Comput. 3, pp. 326-326 (1 page)

Online Publication Date: July 13, 2006

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