Randomized Speedup of the Bellman-Ford Algorithm

We describe a variant of the Bellman-Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm. The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability. We also use our high probability bound to add negative cycle detection to the randomized algorithm.Comment: 12 Pages, 6 Figures, ANALCO 201

Non-adaptive Bellman-Ford: Yen's improvement is optimal

The Bellman-Ford algorithm for single-source shortest paths repeatedly updates tentative distances in an operation called relaxing an edge. In several important applications a non-adaptive (oblivious) implementation is preferred, which means fixing the entire sequence of relaxations upfront, independent of the edge-weights. In a dense graph on $n$ vertices, the algorithm in its standard form performs $(1 + o(1))n^3$ relaxations. An improvement by Yen from 1970 reduces the number of relaxations by a factor of two. We show that no further constant-factor improvements are possible, and every non-adaptive deterministic algorithm based on relaxations must perform $(\frac{1}{2} - o(1))n^3$ steps. This improves an earlier lower bound of Eppstein of $(\frac{1}{6} - o(1))n^3$. Given that a non-adaptive randomized variant of Bellman-Ford with at most $(\frac{1}{3} + o(1))n^3$ relaxations (with high probability) is known, our result implies a strict separation between deterministic and randomized strategies, answering an open question of Eppstein

IMPLEMENTASI BELLMAN-FORD DAN FLOYD-WARSHALL DALAM MENENTUKAN JALUR TERPENDEK MENUJU UNIVERSITAS NASIONAL BERBASIS ANDROID

Disekitar Universitas Nasional memiliki berbagai macam jenis transportasi umum. Sebagian besar masyarakat dan mahasiswa Universitas Nasional masih menggunakan transportasi umum seperti kereta api dan transjakarta, namun masih terkendala jarak antara stasiun dan halte ke Universitas Nasional. Penelitian ini menerapkan algoritma Bellman-Ford dan Floyd-Warshall yang dinilai efektif dan telah banyak digunakan pada penelitian sebelumnya dalam pencarian jalur terpendek diantaranya yaitu pengantaran barang, pencarian kampus dan pencarian lokasi travel. Penelitian ini bertujuan untuk mempermudah masyarakat, khususnya mahasiswa atau peserta didik baru dalam mencari jalur terdekat dari stasiun dan halte menuju Universitas Nasional. Penelitian ini dirancang menggunakan framework flutter dan bahasa pemrograman dart berbasis android dengan pengguna terbanyak pada saat ini. Berdasarkan hasil pengujian, algoritma Bellman-Ford dan Floyd-Warshall untuk kasus pencarian jarak terpendek dari stasiun Pasar Minggu diperoleh jarak terpendek sebesar 1.54 km dengan tingkat keefektifan jarak sebesar 39.40%, sedangkan kasus pencarian jarak terpendek dari halte Jatipadang diperoleh jarak terpendek sebesar 1.97 km dengan tingkat keefektifan jarak sebesar 25.24%

A central limit theorem for temporally non-homogenous Markov chains with applications to dynamic programming

We prove a central limit theorem for a class of additive processes that arise naturally in the theory of finite horizon Markov decision problems. The main theorem generalizes a classic result of Dobrushin (1956) for temporally non-homogeneous Markov chains, and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain. We show through several examples that this added flexibility gives one a direct path to asymptotic normality of the optimal total reward of finite horizon Markov decision problems. The same examples also explain why such results are not easily obtained by alternative Markovian techniques such as enlargement of the state space.Comment: 27 pages, 1 figur

Optimal Online Selection of an Alternating Subsequence: A Central Limit Theorem

We analyze the optimal policy for the sequential selection of an alternating subsequence from a sequence of n independent observations from a continuous distribution F, and we prove a central limit theorem for the number of selections made by that policy. The proof exploits the backward recursion of dynamic programming and assembles a detailed understanding of the associated value functions and selection rules