{"id":7453,"date":"2022-10-21T06:00:38","date_gmt":"2022-10-21T04:00:38","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7453"},"modified":"2022-12-14T12:27:16","modified_gmt":"2022-12-14T10:27:16","slug":"representacion-densa-de-polinomios","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/representacion-densa-de-polinomios\/","title":{"rendered":"Representaci\u00f3n densa de polinomios"},"content":{"rendered":"

Los polinomios pueden representarse de forma dispersa o densa. Por ejemplo, el polinomio 6x^4-5x^2+4x-7 se puede representar de forma dispersa por [6,0,-5,4,-7] y de forma densa por [(4,6),(2,-5),(1,4),(0,-7)].<\/p>\n

Definir la funci\u00f3n<\/p>\n

\n   densa :: [Int] -> [(Int,Int)]\n<\/pre>\n
tal que densa xs<\/code> es la representaci\u00f3n densa del polinomio cuya representaci\u00f3n dispersa es xs<\/code>. Por ejemplo,<\/p>\n
\n   densa [6,0,-5,4,-7]  ==  [(4,6),(2,-5),(1,4),(0,-7)]\n   densa [6,0,0,3,0,4]  ==  [(5,6),(2,3),(0,4)]\n   densa [0]            ==  [(0,0)]\n<\/pre>\n
Soluciones<\/b><\/p>\n
A continuaci\u00f3n se muestran las soluciones en Haskell<\/a> y las soluciones en Python<\/a>.<\/p>\n
<\/a>
\nSoluciones en Haskell<\/b><\/p>\n
\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\ndensa1 :: [Int] -> [(Int,Int)]\ndensa1 xs =\n  [(x,y) | (x,y) <- zip [n-1,n-2..1] xs, y \/= 0]\n  ++ [(0, last xs)]\n  where n = length xs\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\ndensa2 :: [Int] -> [(Int,Int)]\ndensa2 xs =\n  filter (\\ (_,y) -> y \/= 0) (zip [n-1,n-2..1] xs)\n  ++ [(0, last xs)]\n  where n = length xs\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\ndensa3 :: [Int] -> [(Int,Int)]\ndensa3 xs = filter ((\/= 0) . snd) (zip [n-1,n-2..1] xs)\n  ++ [(0, last xs)]\n  where n = length xs\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\ndensa4 :: [Int] -> [(Int,Int)]\ndensa4 xs = aux xs (length xs - 1)\n  where aux [y] 0 = [(0, y)]\n        aux (y:ys) n | y == 0    = aux ys (n-1)\n                     | otherwise = (n,y) : aux ys (n-1)\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_densa :: NonEmptyList Int -> Bool\nprop_densa (NonEmpty xs) =\n  all (== densa1 xs)\n      [densa2 xs,\n       densa3 xs,\n       densa4 xs]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_densa\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> last (densa1 [1..2*10^6])\n--    (0,2000000)\n--    (0.95 secs, 880,569,400 bytes)\n--    \u03bb> last (densa2 [1..2*10^6])\n--    (0,2000000)\n--    (0.52 secs, 800,569,432 bytes)\n--    \u03bb> last (densa3 [1..2*10^6])\n--    (0,2000000)\n--    (0.53 secs, 752,569,552 bytes)\n--    \u03bb> last (densa4 [1..2*10^6])\n--    (0,2000000)\n--    (3.05 secs, 1,267,842,032 bytes)\n--\n--    \u03bb> last (densa1 [1..10^7])\n--    (0,10000000)\n--    (5.43 secs, 4,400,570,128 bytes)\n--    \u03bb> last (densa2 [1..10^7])\n--    (0,10000000)\n--    (3.03 secs, 4,000,570,160 bytes)\n--    \u03bb> last (densa3 [1..10^7])\n--    (0,10000000)\n--    (2.34 secs, 3,760,570,280 bytes)\n<\/pre>\n
El c\u00f3digo se encuentra en GitHub<\/a>.<\/p>\n
<\/a>
\nSoluciones en Python<\/b><\/p>\n
\nfrom sys import setrecursionlimit\nfrom timeit import Timer, default_timer\n\nfrom hypothesis import given\nfrom hypothesis import strategies as st\n\nsetrecursionlimit(10**6)\n\n# 1\u00aa soluci\u00f3n\n# ===========\n\ndef densa1(xs: list[int]) -> list[tuple[int, int]]:\n    n = len(xs)\n    return [(x, y)\n            for (x, y) in zip(range(n-1, 0, -1), xs)\n            if y != 0] + [(0, xs[-1])]\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef densa2(xs: list[int]) -> list[tuple[int, int]]:\n    n = len(xs)\n    return list(filter(lambda p: p[1] != 0,\n                       zip(range(n-1, 0, -1), xs))) + [(0, xs[-1])]\n\n# 3\u00aa soluci\u00f3n\n# ===========\n\ndef densa3(xs: list[int]) -> list[tuple[int, int]]:\n    def aux(ys: list[int], n: int) -> list[tuple[int, int]]:\n        if n == 0:\n            return [(0, ys[0])]\n        if ys[0] == 0:\n            return aux(ys[1:], n-1)\n        return [(n, ys[0])] + aux(ys[1:], n-1)\n\n    return aux(xs, len(xs) - 1)\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\n@given(st.lists(st.integers(), min_size=1))\ndef test_densa(xs: list[int]) -> None:\n    r = densa1(xs)\n    assert densa2(xs) == r\n    assert densa3(xs) == r\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q representacion_densa_de_polinomios.py\n#    1 passed in 0.27s\n\n# Comparaci\u00f3n de eficiencia\n# =========================\n\ndef tiempo(e: str) -> None:\n    \"\"\"Tiempo (en segundos) de evaluar la expresi\u00f3n e.\"\"\"\n    t = Timer(e, \"\", default_timer, globals()).timeit(1)\n    print(f\"{t:0.2f} segundos\")\n\n# La comparaci\u00f3n es\n#    >>> tiempo('densa1(range(1, 10**4))')\n#    0.00 segundos\n#    >>> tiempo('densa2(range(1, 10**4))')\n#    0.00 segundos\n#    >>> tiempo('densa3(range(1, 10**4))')\n#    0.25 segundos\n#\n#    >>> tiempo('densa1(range(1, 10**7))')\n#    1.87 segundos\n#    >>> tiempo('densa2(range(1, 10**7))')\n#    2.15 segundos\n<\/pre>\n
El c\u00f3digo se encuentra en GitHub<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
Los polinomios pueden representarse de forma dispersa o densa. Por ejemplo, el polinomio 6x^4-5x^2+4x-7 se puede representar de forma dispersa por [6,0,-5,4,-7] y de forma densa por [(4,6),(2,-5),(1,4),(0,-7)]. Definir la funci\u00f3n densa :: [Int] -> [(Int,Int)] tal que densa xs es la representaci\u00f3n densa del polinomio cuya representaci\u00f3n dispersa es xs. Por ejemplo, densa [6,0,-5,4,-7]…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[581],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7453"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=7453"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7453\/revisions"}],"predecessor-version":[{"id":7665,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7453\/revisions\/7665"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7453"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7453"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7453"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}