X-Git-Url: https://git.njae.me.uk/?a=blobdiff_plain;f=algorithms.html;h=05309f17fdb55bfecc4eb7626d848f12953f4e83;hb=refs%2Fheads%2Fmaster;hp=b7a7b61c7aea6af1128c18b6b0a91e96bf75b04a;hpb=4a03195ad47aa91d9768584f688c36bb143fc610;p=cas-master-teacher-training.git diff --git a/algorithms.html b/algorithms.html index b7a7b61..05309f1 100644 --- a/algorithms.html +++ b/algorithms.html @@ -50,15 +50,17 @@

-# Algorithms
+# Algorithms ![Open University logo](oulogo_hor.png)
 
 ## What's all the fuss about?
 
+![Alan Turing](alan-turing.jpg)
+
 ---
 
 layout: true
 
-.indexlink[[Index](index.html)]
+.indexlink[![Open University logo](oulogo_hor.png) [Index](index.html)]
 
 ---
 
@@ -71,6 +73,7 @@ Back when computers were human...
 * As much notepaper as you wanted
 * Input â output
 
+.float-right[![right-aligned Growth rates ](turing-machine.jpg)]
 ## Turing machines
 
 Simplified human computer.
@@ -89,6 +92,14 @@ Algorithm is the instructions
 
 Repetition == iteration or recursion
 
+What questions can we ask about algorithms?
+
+* Is it correct?
+* Does it terminate?
+* How long does it take?
+
+Different hardware works at different rates, so ask about how the run time changes with input size.
+
 ---
 
 # Growth rates
@@ -143,6 +154,9 @@ word1 = "cat", word2 = "catty"
 
 # Anagrams version 1a: checking off (fixed)
 
+<table width="100%">
+<tr>
+<td>
 ```
 Given word1, word2
 
@@ -158,6 +172,26 @@ for pointer in range(len(word2)):
         anagram? = False
 return anagram?
 ```
+</td>
+<td>
+```
+Given word1, word2
+
+if len(word1) == len(word2):
+    anagram? = True
+    for character in word1:
+        for pointer in range(len(word2)):
+            if character == word2[pointer]:
+                word2[pointer] = None
+            else:
+                anagram? = False
+    return anagram?
+else:
+    return False
+```
+</td>
+</tr>
+</table>
 
 ---
 
@@ -280,6 +314,14 @@ Given a wordlist and a phrase, find another phrase that's an anagram.
 
 ---
 
+# Sorting with cards
+
+Classroom activity?
+
+Don't forget bogosort!
+
+---
+
 # Dealing with complex problems: some terms to know
 
 (M269 Unit 5)
@@ -292,22 +334,96 @@ Given a wordlist and a phrase, find another phrase that's an anagram.
 * Often leads to logarithmic growth
 
 "Guess the number" game
-    * What if I don't tell you the upper limit?
+
+* What if I don't tell you the upper limit?
 
 ## Dynamic programming
 
 Build a soluion bottom-up from subparts
 
-## Heuristics
+---
+
+# Heuristics
 
 * Posh word for "guess"
 * Greedy algorithms
 * Backtracking
+
+# Parallelism vs concurrency
+
+* Parallelism is implementation
+    * spread the load
+* Concurrency is domain
+    * new task arrives before you've finished this one
+
 ---
 
-# Sorting with cards
+# Making change
 
-Classroom activity?
+Given
+
+* some money and a set of coin denominations,
+
+find
+
+* the smallest number of coins to make that total.
+
+## Approach 1: greedy
+
+```
+Repeat:
+   Pick the biggest coin you can
+```
+
+Only works in some cases
+
+(What the best way to make 7p if you have 1p, 3p, 4p, 5p coins?)
+
+---
+
+# Making change 2: exhaustive search
+
+.float-right[![Ticket to Ride](make-change.dot.png)]
+
+Make a tree of change-making.
+
+Pick the optimal.
+
+---
+
+# Making change 3: dynamic programming
+
+Bottom-up approach
+
+Initial:
+
+0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ...
+-|-|-|-|-|-|-|-|-|-|-|-
+(0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | ...
+
+--
+----
+0* | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ...
+-|-|-|-|-|-|-|-|-|-|-|-
+(0, 0, 0) | (1, 0, 0) | (0, 1, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 1) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | ...
+
+--
+----
+0 | 1* | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ...
+-|-|-|-|-|-|-|-|-|-|-|-
+(0, 0, 0) | (1, 0, 0) | (0, 1, 0) | (1, 1, 0) | (0, 0, 0) | (0, 0, 1) | (1, 0, 1) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | ...
+
+--
+----
+0 | 1 | 2* | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ...
+-|-|-|-|-|-|-|-|-|-|-|-
+(0, 0, 0) | (1, 0, 0) | (0, 1, 0) | (1, 1, 0) | (0, 2, 0) | (0, 0, 1) | (1, 0, 1) | (0, 1, 1) | (0, 0, 0) | (0, 0, 0) | (0, 0, 0) | ...
+
+--
+----
+0 | 1 | 2 | 3* | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ...
+-|-|-|-|-|-|-|-|-|-|-|-
+(0, 0, 0) | (1, 0, 0) | (0, 1, 0) | (1, 1, 0) | (0, 2, 0) | (0, 0, 1) | (1, 0, 1) | (0, 1, 1) | (1, 1, 1) | (0, 0, 0) | (0, 0, 0) | ...
 
 ---
 
@@ -315,19 +431,21 @@ Classroom activity?
 
 ## Travelling Salesman Problem
 
-Online: http://www.math.uwaterloo.ca/tsp/games/tspOnePlayer.html
+.float-right[![Ticket to Ride](ticket-to-ride-small.jpg)]
+
+Online: 
+
+http://www.math.uwaterloo.ca/tsp/games/tspOnePlayer.html
 
 * But many implementations are Java applets, now a problem with security settings
 
 Try Android/iOS apps?
 
-## Change-finding problem
-
-Same as the Knapsack problem: M269 Unit 5 Section 3.2, especially Activity 5.9
+## Ticket to Ride boardgame
 
-Dynamic programming to find solution
+Graph algorithms
 
-## Ticket to Ride boardgame
+* Find shortest path
 
 ---
 
@@ -345,6 +463,14 @@ The computer would either answer "Yes," "No," or never stop working.
 
 This is the Halting problem.
 
+```python
+while i != 0:
+    print(i)
+    i -= 1
+```
+
+Will this halt? When will it not?
+
 ---
 
 # Universal Turing machine
@@ -369,7 +495,141 @@ A *universal* Turing machine can take the description of any Turing machine and
 * Virtual machines
 * ...
 
+---
+
+# Back to halting
+
+* Universal Turing machine can do any possible computation
+* Has the same halting behaviour of the emulated machine
+
+Can we build another machine that detects if a machine halts when given some input?
+
+Assume we can...
+
+```python
+def does_it_stop(program, input):
+    if very_clever_decision:
+        return True
+    else:
+        return False
+```
+
+But I can use the program listing as an input:
+
+```python
+def stops_on_self(program):
+    return does_it_stop(program, program)
+```
+
+---
+
+# Halting: the clever bit
+
+<table width="100%">
+<tr>
+<td>
+```python
+def does_it_stop(program, input):
+    if very_clever_decision:
+        return True
+    else:
+        return False
+```
+</td>
+<td>
+```python
+def stops_on_self(program):
+    return does_it_stop(program, program)
+```
+</td>
+</tr>
+</table>
+
+Let's put an infinite loop in a program that detects infinite loops!
+
+```python
+def bobs_yer_uncle(program):
+    if stops_on_self(program):
+        while True:
+            pass
+    else:
+        return True
+```
+
+If a `program` halts on on itself, `bobs_yer_uncle` doesn't halt. If `program` doesn't halt, `bobs_yer_uncle` returns `True`.
+
+--
+
+What does this do?
+```python
+bobs_yer_uncle(bobs_yer_uncle)
+```
+
+---
+
+<table width="100%">
+<tr>
+<td>
+```python
+def does_it_stop(program, input):
+    if very_clever_decision:
+        return True
+    else:
+        return False
+```
+</td>
+<td>
+```python
+def stops_on_self(program):
+    return does_it_stop(program, program)
+```
+</td>
+<td>
+```python
+def bobs_yer_uncle(program):
+    if stops_on_self(program):
+        while True:
+            pass
+    else:
+        return True
+```
+</td>
+</tr>
+</table>
+
+What does this do?
+```python
+bobs_yer_uncle(bobs_yer_uncle)
+```
 
+<table>
+<tr>
+<th>If it halts...</th>
+<th>If doesn't halt...</th>
+</td>
+<tr>
+<td>
+<ul>
+<li>`stops_on_self(bobs_yer_uncle)` returns `False`</li>
+<li>`does_it_stop(bobs_yer_uncle, bobs_yer_uncle)` returns `False`</li>
+<li>`bobs_yer_uncle(bobs_yer_uncle)` must loop forever</li>
+</ul>
+<b>Contradiction!</b>
+</td>
+<td>
+<ul>
+<li>`stops_on_self(bobs_yer_uncle)` returns `True`</li>
+<li>`does_it_stop(bobs_yer_uncle, bobs_yer_uncle)` returns `True`</li>
+<li>`bobs_yer_uncle(bobs_yer_uncle)` must halt</li>
+</ul>
+<b>Contradiction!</b>
+</td>
+</tr>
+</table>
+
+No flaw in the logic. The only place there could be a mistake is our assumption that `does_it_stop` is possible.
+
+(M269 has a different proof based on different sizes of infinity, and showing that there are infinitely more problems than there are Turing machines to solve them.)