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A property of prefix codes called strong monotonicity is introduced. Then it is proven that for a prefix code C for a given probability distribution, the following are equivalent: (i) C is expected length minimal; (ii) C is length equivalent to a Huffman code; and (iii) C is complete and strongly monotone. Also, three relations are introduced between prefix code trees called same-parent, same-row, and same-probability swap equivalence, and it is shown that for a given source, all Huffman codes are same-parent, same-probability swap equivalent, and all expected length minimal prefix codes are same-row, same-probability swap equivalent.

No abstract

We prove computational intractability of variants of checkers: (1) deciding whether there is a move that forces the other player to win in one move is NP-complete; (2) checkers where players must always be able to jump on their turn is PSPACE-complete; and (3) cooperative versions of (1) and (2) are NP-complete. We also give cooperative checkers puzzles whose solutions are the letters of the alphabet.

No abstract