From c87aafd8744e58fdad2ed402700d80be8ffd0f50 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Ondra=20Mi=C4=8Dka=20=40=20miles-teg?=
<mitch.ondra@gmail.com>
Date: Mon, 30 Mar 2020 14:42:54 +0200
Subject: [PATCH] Fibonacci experiment
---
06-fibonacci_experiment/cpp/Makefile | 22 ++
.../cpp/fibonacci_experiment.cpp | 256 ++++++++++++++++++
06-fibonacci_experiment/cpp/random.h | 61 +++++
06-fibonacci_experiment/python/Makefile | 13 +
.../python/fibonacci_experiment.py | 172 ++++++++++++
06-fibonacci_experiment/task.md | 93 +++++++
6 files changed, 617 insertions(+)
create mode 100644 06-fibonacci_experiment/cpp/Makefile
create mode 100644 06-fibonacci_experiment/cpp/fibonacci_experiment.cpp
create mode 100644 06-fibonacci_experiment/cpp/random.h
create mode 100644 06-fibonacci_experiment/python/Makefile
create mode 100644 06-fibonacci_experiment/python/fibonacci_experiment.py
create mode 100644 06-fibonacci_experiment/task.md
diff --git a/06-fibonacci_experiment/cpp/Makefile b/06-fibonacci_experiment/cpp/Makefile
new file mode 100644
index 0000000..589ab7c
--- /dev/null
+++ b/06-fibonacci_experiment/cpp/Makefile
@@ -0,0 +1,22 @@
+.PHONY: test
+test: fibonacci_experiment
+ @for exp in star random; do \
+ for impl in std naive ; do \
+ echo "t-$$exp-$$impl" ; \
+ time ./fibonacci_experiment $$exp 42 $$impl >t-$$exp-$$impl ; \
+ done ; \
+ done
+ @for b in biased-10 biased-40 biased-70 biased-100 ; do \
+ echo "t-$$b" ; \
+ time ./fibonacci_experiment $$b 42 std >t-$$b ; \
+ done
+
+INCLUDE ?= .
+CXXFLAGS=-std=c++11 -O2 -Wall -Wextra -g -Wno-sign-compare -I$(INCLUDE)
+
+fibonacci_experiment: fibonacci.h fibonacci_experiment.cpp $(INCLUDE)/random.h
+ $(CXX) $(CPPFLAGS) $(CXXFLAGS) fibonacci_experiment.cpp -o $@
+
+.PHONY: clean
+clean:
+ rm -f fibonacci_experiment
diff --git a/06-fibonacci_experiment/cpp/fibonacci_experiment.cpp b/06-fibonacci_experiment/cpp/fibonacci_experiment.cpp
new file mode 100644
index 0000000..3e8d9b5
--- /dev/null
+++ b/06-fibonacci_experiment/cpp/fibonacci_experiment.cpp
@@ -0,0 +1,256 @@
+#include <algorithm>
+#include <functional>
+#include <string>
+#include <utility>
+#include <vector>
+#include <iostream>
+#include <cmath>
+
+#include "fibonacci.h"
+#include "random.h"
+
+void expect_failed(const std::string& message) {
+ fprintf(stderr, "Test error: %s\n", message.c_str());
+ exit(1);
+}
+
+
+/*
+ * A modified Fibonacci heap for benchmarking.
+ *
+ * We inherit the implementation of operations from the FibonacciHeap class
+ * and extend it by keeping statistics on the number of operations
+ * and the total number of structural changes. Also, if naive is turned on,
+ * the decrease does not mark parent which just lost a child.
+ */
+
+class HeapTests : public FibonacciHeap {
+ int ops = 0;
+ int steps = 0;
+ bool naive;
+ size_t max_size = 0;
+
+ public:
+ HeapTests(bool naive = false) : naive(naive) {}
+
+ void stats() {
+ printf(" %.3lf\n", (double)(steps * 1.0 / std::max(1, ops)));
+ }
+
+ Node *insert(priority_t prio, payload_t payload = payload_t()) {
+ ops++;
+ return FibonacciHeap::insert(prio, payload);
+ }
+
+ std::pair<payload_t, priority_t> extract_min() {
+ ops++;
+ return FibonacciHeap::extract_min();
+ }
+
+ void decrease(Node *node, priority_t new_prio) {
+ ops++;
+ if (naive) {
+ EXPECT(node, "Cannot decrease null pointer.");
+ EXPECT(node->prio >= new_prio, "Decrase: new priority larger than old one.");
+
+ node->prio = new_prio;
+ if (is_root(node) || new_prio >= node->parent->prio) return;
+
+ add_child(&meta_root, remove(node));
+ } else FibonacciHeap::decrease(node, new_prio);
+ }
+
+ protected:
+ size_t max_rank() override {
+ max_size = std::max(max_size, size);
+ if (naive) return sqrt(2*max_size) + 2;
+ else return FibonacciHeap::max_rank();
+ }
+
+ void add_child(Node* parent, Node *node) override {
+ steps++;
+ FibonacciHeap::add_child(parent, node);
+ }
+
+ Node *remove(Node *n) override {
+ steps++;
+ return FibonacciHeap::remove(n);
+ }
+};
+
+using Heap = HeapTests;
+
+RandomGen *rng; // Random generator object
+
+
+std::vector<int> log_range(int b, int e, int n) {
+ std::vector<int> ret;
+ for (int i = 0; i < n; i++)
+ ret.push_back(b * exp(log(e / b) / (n - 1) * i));
+ return ret;
+}
+
+// An auxiliary function for generating a random permutation.
+template < typename Vector >
+void random_permutation(Vector& v)
+{
+ for (int i = 0; i < v.size(); i++)
+ std::swap(v[i], v[i + rng->next_range(v.size() - i)]);
+}
+
+// Remove given node from heap.
+void remove(Heap* H, Node *n) {
+ H->decrease(n, -1000*1000*1000);
+ H->extract_min();
+}
+
+// Force consolidation of the heap.
+void consolidate(Heap* H) {
+ remove(H, H->insert(0));
+}
+
+// Construct a star with n nodes and root index r.
+void star(Heap *H, Node **node_map, int n, int r, bool consolidate_) {
+ if (n == 1) {
+ EXPECT(!node_map[r], "");
+ node_map[r] = H->insert(r);
+ if (consolidate_) consolidate(H);
+ } else {
+ star(H, node_map, n - 1, r, false);
+ star(H, node_map, n - 1, r + n - 1, true);
+
+ for (int i = r + n; i < r + 2*n - 2; i++) {
+ remove(H, node_map[i]);
+ node_map[i] = nullptr;
+ }
+ }
+}
+
+/* Generates a sequence on which non-cascading heaps need lots of time.
+ * Source: "Replacing Mark Bits with Randomness in Fibonacci Heaps" Jerry Li and John Peebles, MIT
+ * -> modified so that only a quadratic number of elements are needed in the star construction.
+ */
+void test_star(bool naive) {
+ for (int i = 1; i < 17; i++) {
+ int start = 3;
+ int N = start + i * (i+1) / 2;
+ Heap H(naive);
+ Node **node_map = new Node*[N];
+ for (int i = 0; i < N; i++) node_map[i] = nullptr;
+
+ for (int j = i; j > 0; j--) {
+ star(&H, node_map, j, start, false);
+ start += j;
+ }
+
+ for (int j = 0; j < (1 << i); j++) {
+ H.insert(1);
+ H.insert(1);
+ H.extract_min();
+ H.extract_min();
+ }
+
+ printf("%i", N);
+ H.stats();
+
+ delete[] node_map;
+ }
+}
+
+/* A random test.
+ *
+ * The test does 2N insertions first, and then N random operations insert / decrease /
+ * extract_min, each with probability proprotional to its weight ins / dec / rem.
+ */
+void test_random(int ins, int dec, int rem, bool naive) {
+ for (int N : log_range(50, 80000, 30)) {
+ Heap H(naive);
+ std::vector<Node*> node_map;
+ int s = ins + dec + rem;
+
+ std::vector<int> ops;
+ for (int i = 0; i < (N*ins) / s; i++) ops.push_back(0);
+ for (int i = 0; i < (N*dec) / s; i++) ops.push_back(1);
+ for (int i = 0; i < (N*rem) / s; i++) ops.push_back(2);
+ random_permutation(ops);
+
+# define INSERT() node_map.push_back(H.insert(rng->next_range(5*N), node_map.size()))
+
+ for (int i = 0; i < 2*N; i++) INSERT();
+
+ for (int op : ops) switch (op) {
+ case 0:
+ INSERT();
+ break;
+ case 1: {
+ Node *node = node_map[rng->next_range(node_map.size())];
+ int p = node->priority() - rng->next_range(N / 5) - 1;
+ H.decrease(node, p);
+ break;
+ }
+ default: {
+ auto ret = H.extract_min();
+ if (ret.first + 1 != node_map.size()) {
+ node_map[ret.first] = node_map.back();
+ node_map[ret.first]->payload = ret.first;
+ }
+ node_map.pop_back();
+ }
+ }
+
+ printf("%i", N);
+ H.stats();
+
+# undef INSERT
+ }
+}
+
+
+std::vector<std::pair<std::string, std::function<void(bool)>>> tests = {
+ { "star", test_star },
+ { "random", [](bool n){ test_random(10, 10, 10, n); } },
+ { "biased-10", [](bool n){ test_random(10, 10, 20, n); } },
+ { "biased-40", [](bool n){ test_random(10, 40, 20, n); } },
+ { "biased-70", [](bool n){ test_random(10, 70, 20, n); } },
+ { "biased-100", [](bool n){ test_random(10, 100, 20, n); } },
+};
+
+int main(int argc, char **argv)
+{
+ if (argc != 4) {
+ fprintf(stderr, "Usage: %s <test> <student-id> (std|naive)\n", argv[0]);
+ return 1;
+ }
+
+ std::string which_test = argv[1];
+ std::string id_str = argv[2];
+ std::string mode = argv[3];
+
+ try {
+ rng = new RandomGen(stoi(id_str));
+ } catch (...) {
+ fprintf(stderr, "Invalid student ID\n");
+ return 1;
+ }
+
+ bool naive;
+ if (mode == "std")
+ naive = false;
+ else if (mode == "naive")
+ naive = true;
+ else
+ {
+ fprintf(stderr, "Last argument must be either 'std' or 'naive'\n");
+ return 1;
+ }
+
+ for (const auto& test : tests) {
+ if (test.first == which_test) {
+ test.second(naive);
+ return 0;
+ }
+ }
+
+ fprintf(stderr, "Unknown test %s\n", which_test.c_str());
+ return 1;
+}
diff --git a/06-fibonacci_experiment/cpp/random.h b/06-fibonacci_experiment/cpp/random.h
new file mode 100644
index 0000000..5ef10ae
--- /dev/null
+++ b/06-fibonacci_experiment/cpp/random.h
@@ -0,0 +1,61 @@
+#ifndef DS1_RANDOM_H
+#define DS1_RANDOM_H
+
+#include <cstdint>
+
+/*
+ * This is the xoroshiro128+ random generator, designed in 2016 by David Blackman
+ * and Sebastiano Vigna, distributed under the CC-0 license. For more details,
+ * see http://vigna.di.unimi.it/xorshift/.
+ *
+ * Rewritten to C++ by Martin Mares, also placed under CC-0.
+ */
+
+class RandomGen {
+ uint64_t state[2];
+
+ uint64_t rotl(uint64_t x, int k)
+ {
+ return (x << k) | (x >> (64 - k));
+ }
+
+ public:
+ // Initialize the generator, set its seed and warm it up.
+ RandomGen(unsigned int seed)
+ {
+ state[0] = seed * 0xdeadbeef;
+ state[1] = seed ^ 0xc0de1234;
+ for (int i=0; i<100; i++)
+ next_u64();
+ }
+
+ // Generate a random 64-bit number.
+ uint64_t next_u64(void)
+ {
+ uint64_t s0 = state[0], s1 = state[1];
+ uint64_t result = s0 + s1;
+ s1 ^= s0;
+ state[0] = rotl(s0, 55) ^ s1 ^ (s1 << 14);
+ state[1] = rotl(s1, 36);
+ return result;
+ }
+
+ // Generate a random 32-bit number.
+ uint32_t next_u32(void)
+ {
+ return next_u64() >> 11;
+ }
+
+ // Generate a number between 0 and range-1.
+ unsigned int next_range(unsigned int range)
+ {
+ /*
+ * This is not perfectly uniform, unless the range is a power of two.
+ * However, for 64-bit random values and 32-bit ranges, the bias is
+ * insignificant.
+ */
+ return next_u64() % range;
+ }
+};
+
+#endif
diff --git a/06-fibonacci_experiment/python/Makefile b/06-fibonacci_experiment/python/Makefile
new file mode 100644
index 0000000..656f937
--- /dev/null
+++ b/06-fibonacci_experiment/python/Makefile
@@ -0,0 +1,13 @@
+.PHONY: plot
+test:
+ @for exp in star random ; do \
+ for impl in std naive ; do \
+ echo "t-$$exp-$$impl" ; \
+ time ./fibonacci_experiment.py $$exp 42 $$impl >t-$$exp-$$impl ; \
+ done ; \
+ done
+ @for b in biased-10 biased-40 biased-70 biased-100 ; do \
+ echo "t-$$b" ; \
+ time ./fibonacci_experiment.py $$b 42 std >t-$$b ; \
+ done
+
diff --git a/06-fibonacci_experiment/python/fibonacci_experiment.py b/06-fibonacci_experiment/python/fibonacci_experiment.py
new file mode 100644
index 0000000..eb7a091
--- /dev/null
+++ b/06-fibonacci_experiment/python/fibonacci_experiment.py
@@ -0,0 +1,172 @@
+#!/usr/bin/env python3
+
+import sys, itertools, random
+from math import sqrt, log, exp
+from fibonacci import FibonacciHeap as Heap
+
+class BenchmarkingHeap(Heap):
+ """A modified Fibonacci heap for benchmarking.
+
+ We inherit the implementation of operations from the FibonacciHeap class
+ and extend it by keeping statistics on the number of operations
+ and the total number of structural changes. Also, if naive is turned on,
+ the decrease does not mark parent which just lost a child.
+ """
+
+ def __init__(self, naive=False):
+ Heap.__init__(self)
+ self.naive = naive
+ self._ops = 0
+ self._steps = 0
+ self._max_size = 0 # used to get upper bound on number of buckets in naive version
+
+ def stats(self):
+ return " %.3f" % (self._steps / max(self._ops, 1))
+
+ def insert(self, *args):
+ self._ops += 1
+ return Heap.insert(self, *args)
+
+ def decrease(self, node, new_prio):
+ self._ops += 1
+ if self.naive:
+ assert node is not None, "Cannot decrease None."
+ assert node is not self._meta_root, "Cannot decrease meta root."
+ assert new_prio <= node._prio, \
+ "Decrease: new priority is bigger than old one."
+
+ node._prio = new_prio
+ if self._is_root(node) or node._prio >= node._parent._prio:
+ return
+
+ self._add_child(self._meta_root, self._remove(node))
+ else:
+ Heap.decrease(self, node, new_prio)
+
+ def extract_min(self):
+ self._ops += 1
+ return Heap.extract_min(self)
+
+ def _add_child(self, *args):
+ self._steps += 1
+ Heap._add_child(self, *args)
+
+ def _remove(self, *args):
+ self._steps += 1
+ return Heap._remove(self, *args)
+
+ def _max_rank(self):
+ self._max_size = max(self._max_size, self._size)
+ if self.naive: return int(sqrt(2 * self._max_size)) + 2
+ return Heap._max_rank(self)
+
+def remove(H, node):
+ H.decrease(node, -10**9)
+ H.extract_min()
+
+def consolidate(H):
+ remove(H, H.insert(0))
+
+def log_range(b, e, n):
+ assert 0 < b < e
+ for i in range(n):
+ yield int(b * exp(log(e / b) / (n - 1) * i))
+
+def star(H, node_map, n, r, consolidate_):
+ """Construct a star with n nodes and root index r.
+ """
+ if n == 1:
+ assert node_map[r] is None
+ node_map[r] = H.insert(r)
+ if consolidate_:
+ consolidate(H)
+ else:
+ star(H, node_map, n - 1, r, False)
+ star(H, node_map, n - 1, r + n - 1, True)
+
+ for i in range(r + n, r + 2*n - 2):
+ remove(H, node_map[i])
+ node_map[i] = None
+
+def test_star(naive):
+ """Generates a sequence on which non-cascading heaps need lots of time.
+ Source: "Replacing Mark Bits with Randomness in Fibonacci Heaps" Jerry Li and John Peebles, MIT
+ -> modified so that only a quadratic number of elements are needed in the star construction.
+ """
+ for i in range(1, 17): # was 26
+ start = 3
+ N = start + i * (i+1) // 2
+ H = BenchmarkingHeap(naive)
+ node_map = [None] * N
+
+ for j in range(i, 0, -1):
+ star(H, node_map, j, start, False)
+ start += j
+
+ for i in range(1 << i):
+ H.insert(1)
+ H.insert(1)
+ H.extract_min()
+ H.extract_min()
+
+ print(N, H.stats())
+
+def test_random(ins, dec, rem, naive):
+ """A random test.
+
+ The test does 2N insertions first, and then N random operations insert / decrease /
+ extract_min each with probability proprotional to its weight ins / dec / rem.
+ """
+ for N in log_range(50, 80000, 30):
+ H = BenchmarkingHeap(naive)
+ node_map = []
+ s = ins + dec + rem
+ ops = [ 0 ] * ((N*ins) // s) + [ 1 ] * ((N*dec) // s) + [ 2 ] * ((N*rem) // s)
+ random.shuffle(ops)
+
+ def insert():
+ node_map.append(H.insert(random.randrange(5 * N), len(node_map)))
+
+ for _ in range(2*N): insert()
+
+ for op in ops:
+ if op == 0: insert()
+ elif op == 1:
+ node = random.choice(node_map)
+ p = node.priority() - random.randrange(N // 5) - 1
+ H.decrease(node, p)
+ else:
+ i, _ = H.extract_min()
+ if i + 1 == len(node_map):
+ node_map.pop()
+ else:
+ node_map[i] = node_map.pop()
+ node_map[i].payload = i
+
+ print(N, H.stats())
+
+tests = {
+ "star": test_star,
+ "random": lambda n: test_random(10, 10, 10, n),
+ "biased-10": lambda n: test_random(10, 10, 20, n),
+ "biased-40": lambda n: test_random(10, 40, 20, n),
+ "biased-70": lambda n: test_random(10, 70, 20, n),
+ "biased-100": lambda n: test_random(10, 100, 20, n),
+}
+
+if len(sys.argv) == 4:
+ test, student_id = sys.argv[1], sys.argv[2]
+ if sys.argv[3] == "std":
+ naive = False
+ elif sys.argv[3] == "naive":
+ naive = True
+ else:
+ raise ValueError("Last argument must be either 'std' or 'naive'")
+ random.seed(int(student_id))
+ if test in tests:
+ tests[test](naive)
+ else:
+ raise ValueError("Unknown test {}".format(test))
+else:
+ raise ValueError("Usage: {} <test> <student-id> (std|naive)".format(sys.argv[0]))
+
diff --git a/06-fibonacci_experiment/task.md b/06-fibonacci_experiment/task.md
new file mode 100644
index 0000000..4e834ba
--- /dev/null
+++ b/06-fibonacci_experiment/task.md
@@ -0,0 +1,93 @@
+## Goal
+
+The goal of this assignment is to evaluate your implementation of Fibonacci heap
+experimentally and to compare it with a "naive" implementation which does not
+mark vertices after cutting a child.
+
+You are given a test program (`fibonacci_experiment`) which calls your
+implementation from the previous assignment to perform the following
+experiments:
+
+- _Star test:_ Specific sequence of operations on which naive implementation
+ creates $Θ(\sqrt{n})$ stars of size 1 up to $Θ(\sqrt{n})$.
+- _Random test:_ Test of size $n$ first inserts $2n$ elements and then it does
+ $n$ operations of which one third are insertions, one third are decreases
+ and one third are extract\_mins in random order.
+- _Biased tests:_ Like random test but the weights of operations are different.
+ First the test inserts $2n$ elements and then it does random $n$ operations.
+ Number of operations of each type is proportional to their weight.
+ Weight of insert is 10, weight of extract\_min 20 and weight of decrease is
+ the number in the name of the test, and implemented possibilities are 10, 40,
+ 70 and 100.
+
+The program tries each experiment with different values of $n$. In each try,
+it prints the average number of structural changes (adding and removing a child
+of a node) per one operation (insert, decrease and extract\_min).
+
+You should perform these experiments and write a report, which contains one
+plot of the measured data for each test. The plots should be following:
+
+- _Star test:_ One plot with two curves – one for standard and the other one for naive
+ version.
+- _Random test:_ One plot with two curves – one for standard and the other one for naive
+ version.
+- _Biased tests:_ One plot with 4 curves – one for each value of bias, all of them tested
+ with standard implementation. Note that we do not do biased test naive implementation.
+
+Each plot should show the dependence
+of the average number of structural changes on the test size $n$.
+
+The report should discuss the experimental results and try to explain the observed
+behavior using theory from the lectures. (If you want, you can carry out further
+experiments to gain better understanding of the data structure and include these
+in the report. This is strictly optional.)
+
+You should submit a PDF file with the report (and no source code).
+You will get 1 temporary point upon submission if the file is syntactically correct;
+proper points will be assigned later.
+
+## Test program
+
+The test program is given three arguments:
+- The name of the test (`star`, `random`, `biased-10`, `biased-40`, `biased-70`, `biased-100`).
+- The random seed: you should use the last 2 digits of your student ID (you can find
+ it in the Study Information System – just click on the Personal data icon). Please
+ include the random seed in your report.
+- The implementation to test (`std` or `naive`).
+
+The output of the program contains one line per experiment, which consists of
+the set size and the average number of structural changes.
+
+## Your implementation
+
+Please use your implementation from the previous exercise. If you used C++,
+make sure that the `add_child()`,
+`remove()` and `max_rank()` methods are declared virtual (they will
+be overriden by the test program). Also, if you are performing any structural changes
+without using the `add_child()` and `remove()` helper methods, you need to adjust
+the `BenchmarkingHeap` class accordingly.
+
+## Hints
+
+The following tools can be useful for producing nice plots:
+- [pandas](https://pandas.pydata.org/)
+- [matplotlib](https://matplotlib.org/)
+- [gnuplot](http://www.gnuplot.info/)
+
+A quick checklist for plots:
+- Is there a caption explaining what is plotted?
+- Are the axes clearly labelled? Do they have value ranges and units?
+- Have you mentioned that this axis has logarithmic scale? (Logarithmic graphs
+ are more fitting in some cases, but you should tell.)
+- Is it clear which curve means what?
+- Is it clear what are the measured points and what is an interpolated
+ curve between them?
+- Are there any overlaps? (E.g., the most interesting part of the curve
+ hidden underneath a label?)
+
+In your discussion, please distinguish the following kinds of claims.
+It should be always clear which is which:
+- Experimental results (i.e., the raw data you obtained from the experiments)
+- Theoretical facts (i.e., claims we have proved mathematically)
+- Your hypotheses (e.g., when you claim that the graph looks like something is true,
+ but you are not able to prove rigorously that it always holds)
--
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