## Primality test in Haskell

**Problem:** Implement a primality test in Haskell. Below is the code. The first implementation of the function divides uses a homemade function for finding the remainder when an integer is divided by another integer.

-- The remainder when a is divided by b. remainder :: Integer -> Integer -> Integer remainder a b | a < b = a | a == b = 0 | otherwise = remainder (a - b) b -- Whether d divides n. divides :: Integer -> Integer -> Bool divides d n = remainder n d == 0

The second implementation of divides uses the built-in function rem to find the remainder upon division by an integer.

-- Whether d divides n. A more efficient version that uses the built-in -- function rem. divides :: Integer -> Integer -> Bool divides d n = rem n d == 0

The full primality test follows:

-- Whether d divides n. A more efficient version that uses the built-in -- function rem. divides :: Integer -> Integer -> Bool divides d n = rem n d == 0 -- The least divisor of n that is at least k. ldf :: Integer -> Integer -> Integer ldf k n | divides k n = k | k^2 > n = n | otherwise = ldf (k + 1) n -- The least divisor of n. ld :: Integer -> Integer ld n = ldf 2 n -- Primality test. prime :: Integer -> Bool prime n | n < 1 = error "must be a positive integer" | n == 1 = False | otherwise = ld n == n

## Basic implementation of Dijkstra’s algorithm

Dijkstra’s algorithm as presented in Algorithm 2.5, page 75 of the book Algorithmic Graph Theory is meant to be a general template. Lots of details have been left out, one in particular is how to implement line 6 of the algorithm. This one line of Dijkstra’s algorithm has been the subject of numerous research papers on how to efficiently implement a search technique for Dijkstra’s algorithm. A simple search technique is linear search, where you search some array or list from start to finish. A more efficient search technique for line 6 is a binary heap. To implement infinity as stated on line 1 of Algorithm 2.5, you would simply let a very large number represent infinity. This number should ideally be larger than any weight in the graph you are searching.

Below is a basic implementation of Dijkstra’s algorithm following the general template of Algorithm 2.5. This implementation is meant to be for searching in simple, unweighted, undirected, connected graphs. Because the graph to search is assumed to be unweighted, I simply let each edge have unit weight and represent infinity as the integer . The implementation below should provide a basis on which to implement Dijkstra’s algorithm for, say, weighted graphs and other types of graphs. To use the implementation below, save it to a Python file, load the file into a Sage session using load(), and call the function dijkstra().

def dijkstra(G, s): """ Shortest paths in a graph using Dijkstra's algorithm. INPUT: - G -- a simple, unweighted, undirected, connected graph. Thus each edge has unit weight. - s -- a vertex in G from which to start the search. OUTPUT: A list D of distances such that D[v] is the distance of a shortest path from s to v. A dictionary P of vertex parents such that P[v] is the parent of v. EXAMPLES: sage: G = graphs.PetersenGraph() sage: dijkstra(G, 0) ([0, 1, 2, 2, 1, 1, 2, 2, 2, 2], {1: 0, 2: 1, 3: 4, 4: 0, 5: 0, 6: 1, 7: 5, 8: 5, 9: 4}) sage: G = Graph({0:{1:1, 3:1}, 1:{2:1, 3:1, 4:1}, 2:{4:1}}) sage: dijkstra(G, 0) ([0, 1, 2, 1, 2], {1: 0, 2: 1, 3: 0, 4: 1}) """ n = G.order() # how many vertices m = G.size() # how many edges D = [1000000000 for _ in range(n)] # largest weights; represent +infinity D[s] = 0 # distance from vertex to itself is zero P = {} # a dictionary for fast look-up Q = set(G.vertices()) while len(Q) > 0: v = mindist(D, Q) Q.remove(v) Adj = set(G.neighbors(v)) for u in Adj.intersection(Q): if D[u] > D[v] + 1: # each edge has unit weight, so add 1 D[u] = D[v] + 1 P.setdefault(u, v) # the parent of u is v return D, P def mindist(D, Q): """ Choose a vertex in Q such that it has minimal distance. INPUT: - D -- a list of vertices with corresponding distances. Each distance D[v] corresponding to a vertex v means that v is that much further away from a source vertex. - Q -- all vertices to consider. OUTPUT: A vertex with minimum distance. """ v = None # start the search here low = 1000000000 # the running minimum distance; represent +infinity for u in Q: if D[u] < low: v = u low = D[v] return v

## Haskell snippet

Just some snippet of Haskell code today to return the last but one element of a list. The first implementation uses the built-in functions `last`

and `take`

.

-- Return the last but one element of a list. This implementation uses -- last and take. penlast :: [a] -> a penlast xs = if null xs || length xs <= 2 then head xs else last (take ((length xs) - 1) xs)

The second implementation below uses the built-in functions `head`

and `drop`

.

-- Return the penultimate element of a list. This implementation uses -- head and drop. penhead :: [a] -> a penhead xs = if null xs || length xs <= 2 then head xs else head (drop ((length xs) - 2) xs)