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+# util.py
+# -------
+# Licensing Information:  You are free to use or extend these projects for
+# educational purposes provided that (1) you do not distribute or publish
+# solutions, (2) you retain this notice, and (3) you provide clear
+# attribution to UC Berkeley, including a link to http://ai.berkeley.edu.
+# 
+# Attribution Information: The Pacman AI projects were developed at UC Berkeley.
+# The core projects and autograders were primarily created by John DeNero
+# (denero@cs.berkeley.edu) and Dan Klein (klein@cs.berkeley.edu).
+# Student side autograding was added by Brad Miller, Nick Hay, and
+# Pieter Abbeel (pabbeel@cs.berkeley.edu).
+
+
+# util.py
+# -------
+# Licensing Information:  You are free to use or extend these projects for
+# educational purposes provided that (1) you do not distribute or publish
+# solutions, (2) you retain this notice, and (3) you provide clear
+# attribution to UC Berkeley, including a link to http://ai.berkeley.edu.
+# 
+# Attribution Information: The Pacman AI projects were developed at UC Berkeley.
+# The core projects and autograders were primarily created by John DeNero
+# (denero@cs.berkeley.edu) and Dan Klein (klein@cs.berkeley.edu).
+# Student side autograding was added by Brad Miller, Nick Hay, and
+# Pieter Abbeel (pabbeel@cs.berkeley.edu).
+
+
+import sys
+import inspect
+import heapq, random
+import cStringIO
+
+
+class FixedRandom:
+    def __init__(self):
+        fixedState = (3, (2147483648L, 507801126L, 683453281L, 310439348L, 2597246090L, \
+            2209084787L, 2267831527L, 979920060L, 3098657677L, 37650879L, 807947081L, 3974896263L, \
+            881243242L, 3100634921L, 1334775171L, 3965168385L, 746264660L, 4074750168L, 500078808L, \
+            776561771L, 702988163L, 1636311725L, 2559226045L, 157578202L, 2498342920L, 2794591496L, \
+            4130598723L, 496985844L, 2944563015L, 3731321600L, 3514814613L, 3362575829L, 3038768745L, \
+            2206497038L, 1108748846L, 1317460727L, 3134077628L, 988312410L, 1674063516L, 746456451L, \
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+            3875025632L, 2466137587L, 1235845595L, 4214575620L, 3792516855L, 657994358L, 1241843248L, \
+            1695651859L, 3678946666L, 1929922113L, 2351044952L, 2317810202L, 2039319015L, 460787996L, \
+            3654096216L, 4068721415L, 1814163703L, 2904112444L, 1386111013L, 574629867L, 2654529343L, \
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+            3034757648L, 741703672L, 3094328738L, 459332691L, 2702383376L, 1610239915L, 4162939394L, \
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+            2340271949L, 867277230L, 868513116L, 2158535651L, 2487822909L, 3428235761L, 3067196046L, \
+            3435119657L, 1908441839L, 788668797L, 3367703138L, 3317763187L, 908264443L, 2252100381L, \
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+            1253909612L, 1096494446L, 555725445L, 2277045895L, 3340096504L, 1383318686L, 4234428127L, \
+            1072582179L, 94169494L, 1064509968L, 2681151917L, 2681864920L, 734708852L, 1338914021L, \
+            1270409500L, 1789469116L, 4191988204L, 1716329784L, 2213764829L, 3712538840L, 919910444L, \
+            1318414447L, 3383806712L, 3054941722L, 3378649942L, 1205735655L, 1268136494L, 2214009444L, \
+            2532395133L, 3232230447L, 230294038L, 342599089L, 772808141L, 4096882234L, 3146662953L, \
+            2784264306L, 1860954704L, 2675279609L, 2984212876L, 2466966981L, 2627986059L, 2985545332L, \
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+            3199632491L, 2235795585L, 2865407118L, 36763651L, 2441503575L, 3314890374L, 1755526087L, \
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+            647778910L, 275904777L, 1249389189L, 3640887431L, 4200779599L, 323384601L, 3446088641L, \
+            4049835786L, 1718989062L, 3563787136L, 44099190L, 3281263107L, 22910812L, 1826109246L, \
+            745118154L, 3392171319L, 1571490704L, 354891067L, 815955642L, 1453450421L, 940015623L, \
+            796817754L, 1260148619L, 3898237757L, 176670141L, 1870249326L, 3317738680L, 448918002L, \
+            4059166594L, 2003827551L, 987091377L, 224855998L, 3520570137L, 789522610L, 2604445123L, \
+            454472869L, 475688926L, 2990723466L, 523362238L, 3897608102L, 806637149L, 2642229586L, \
+            2928614432L, 1564415411L, 1691381054L, 3816907227L, 4082581003L, 1895544448L, 3728217394L, \
+            3214813157L, 4054301607L, 1882632454L, 2873728645L, 3694943071L, 1297991732L, 2101682438L, \
+            3952579552L, 678650400L, 1391722293L, 478833748L, 2976468591L, 158586606L, 2576499787L, \
+            662690848L, 3799889765L, 3328894692L, 2474578497L, 2383901391L, 1718193504L, 3003184595L, \
+            3630561213L, 1929441113L, 3848238627L, 1594310094L, 3040359840L, 3051803867L, 2462788790L, \
+            954409915L, 802581771L, 681703307L, 545982392L, 2738993819L, 8025358L, 2827719383L, \
+            770471093L, 3484895980L, 3111306320L, 3900000891L, 2116916652L, 397746721L, 2087689510L, \
+            721433935L, 1396088885L, 2751612384L, 1998988613L, 2135074843L, 2521131298L, 707009172L, \
+            2398321482L, 688041159L, 2264560137L, 482388305L, 207864885L, 3735036991L, 3490348331L, \
+            1963642811L, 3260224305L, 3493564223L, 1939428454L, 1128799656L, 1366012432L, 2858822447L, \
+            1428147157L, 2261125391L, 1611208390L, 1134826333L, 2374102525L, 3833625209L, 2266397263L, \
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+            509704467L, 2582695198L, 4256268040L, 3391197562L, 1460642842L, 1617931012L, 457825497L, \
+            1031452907L, 1330422862L, 4125947620L, 2280712485L, 431892090L, 2387410588L, 2061126784L, \
+            896457479L, 3480499461L, 2488196663L, 4021103792L, 1877063114L, 2744470201L, 1046140599L, \
+            2129952955L, 3583049218L, 4217723693L, 2720341743L, 820661843L, 1079873609L, 3360954200L, \
+            3652304997L, 3335838575L, 2178810636L, 1908053374L, 4026721976L, 1793145418L, 476541615L, \
+            973420250L, 515553040L, 919292001L, 2601786155L, 1685119450L, 3030170809L, 1590676150L, \
+            1665099167L, 651151584L, 2077190587L, 957892642L, 646336572L, 2743719258L, 866169074L, \
+            851118829L, 4225766285L, 963748226L, 799549420L, 1955032629L, 799460000L, 2425744063L, \
+            2441291571L, 1928963772L, 528930629L, 2591962884L, 3495142819L, 1896021824L, 901320159L, \
+            3181820243L, 843061941L, 3338628510L, 3782438992L, 9515330L, 1705797226L, 953535929L, \
+            764833876L, 3202464965L, 2970244591L, 519154982L, 3390617541L, 566616744L, 3438031503L, \
+            1853838297L, 170608755L, 1393728434L, 676900116L, 3184965776L, 1843100290L, 78995357L, \
+            2227939888L, 3460264600L, 1745705055L, 1474086965L, 572796246L, 4081303004L, 882828851L, \
+            1295445825L, 137639900L, 3304579600L, 2722437017L, 4093422709L, 273203373L, 2666507854L, \
+            3998836510L, 493829981L, 1623949669L, 3482036755L, 3390023939L, 833233937L, 1639668730L, \
+            1499455075L, 249728260L, 1210694006L, 3836497489L, 1551488720L, 3253074267L, 3388238003L, \
+            2372035079L, 3945715164L, 2029501215L, 3362012634L, 2007375355L, 4074709820L, 631485888L, \
+            3135015769L, 4273087084L, 3648076204L, 2739943601L, 1374020358L, 1760722448L, 3773939706L, \
+            1313027823L, 1895251226L, 4224465911L, 421382535L, 1141067370L, 3660034846L, 3393185650L, \
+            1850995280L, 1451917312L, 3841455409L, 3926840308L, 1397397252L, 2572864479L, 2500171350L, \
+            3119920613L, 531400869L, 1626487579L, 1099320497L, 407414753L, 2438623324L, 99073255L, \
+            3175491512L, 656431560L, 1153671785L, 236307875L, 2824738046L, 2320621382L, 892174056L, \
+            230984053L, 719791226L, 2718891946L, 624L), None)
+        self.random = random.Random()
+        self.random.setstate(fixedState)
+
+"""
+ Data structures useful for implementing SearchAgents
+"""
+
+class Stack:
+    "A container with a last-in-first-out (LIFO) queuing policy."
+    def __init__(self):
+        self.list = []
+
+    def push(self,item):
+        "Push 'item' onto the stack"
+        self.list.append(item)
+
+    def pop(self):
+        "Pop the most recently pushed item from the stack"
+        return self.list.pop()
+
+    def isEmpty(self):
+        "Returns true if the stack is empty"
+        return len(self.list) == 0
+
+class Queue:
+    "A container with a first-in-first-out (FIFO) queuing policy."
+    def __init__(self):
+        self.list = []
+
+    def push(self,item):
+        "Enqueue the 'item' into the queue"
+        self.list.insert(0,item)
+
+    def pop(self):
+        """
+          Dequeue the earliest enqueued item still in the queue. This
+          operation removes the item from the queue.
+        """
+        return self.list.pop()
+
+    def isEmpty(self):
+        "Returns true if the queue is empty"
+        return len(self.list) == 0
+
+class PriorityQueue:
+    """
+      Implements a priority queue data structure. Each inserted item
+      has a priority associated with it and the client is usually interested
+      in quick retrieval of the lowest-priority item in the queue. This
+      data structure allows O(1) access to the lowest-priority item.
+    """
+    def  __init__(self):
+        self.heap = []
+        self.count = 0
+
+    def push(self, item, priority):
+        entry = (priority, self.count, item)
+        heapq.heappush(self.heap, entry)
+        self.count += 1
+
+    def pop(self):
+        (_, _, item) = heapq.heappop(self.heap)
+        return item
+
+    def isEmpty(self):
+        return len(self.heap) == 0
+
+    def update(self, item, priority):
+        # If item already in priority queue with higher priority, update its priority and rebuild the heap.
+        # If item already in priority queue with equal or lower priority, do nothing.
+        # If item not in priority queue, do the same thing as self.push.
+        for index, (p, c, i) in enumerate(self.heap):
+            if i == item:
+                if p <= priority:
+                    break
+                del self.heap[index]
+                self.heap.append((priority, c, item))
+                heapq.heapify(self.heap)
+                break
+        else:
+            self.push(item, priority)
+
+class PriorityQueueWithFunction(PriorityQueue):
+    """
+    Implements a priority queue with the same push/pop signature of the
+    Queue and the Stack classes. This is designed for drop-in replacement for
+    those two classes. The caller has to provide a priority function, which
+    extracts each item's priority.
+    """
+    def  __init__(self, priorityFunction):
+        "priorityFunction (item) -> priority"
+        self.priorityFunction = priorityFunction      # store the priority function
+        PriorityQueue.__init__(self)        # super-class initializer
+
+    def push(self, item):
+        "Adds an item to the queue with priority from the priority function"
+        PriorityQueue.push(self, item, self.priorityFunction(item))
+
+
+def manhattanDistance( xy1, xy2 ):
+    "Returns the Manhattan distance between points xy1 and xy2"
+    return abs( xy1[0] - xy2[0] ) + abs( xy1[1] - xy2[1] )
+
+"""
+  Data structures and functions useful for various course projects
+
+  The search project should not need anything below this line.
+"""
+
+class Counter(dict):
+    """
+    A counter keeps track of counts for a set of keys.
+
+    The counter class is an extension of the standard python
+    dictionary type.  It is specialized to have number values
+    (integers or floats), and includes a handful of additional
+    functions to ease the task of counting data.  In particular,
+    all keys are defaulted to have value 0.  Using a dictionary:
+
+    a = {}
+    print a['test']
+
+    would give an error, while the Counter class analogue:
+
+    >>> a = Counter()
+    >>> print a['test']
+    0
+
+    returns the default 0 value. Note that to reference a key
+    that you know is contained in the counter,
+    you can still use the dictionary syntax:
+
+    >>> a = Counter()
+    >>> a['test'] = 2
+    >>> print a['test']
+    2
+
+    This is very useful for counting things without initializing their counts,
+    see for example:
+
+    >>> a['blah'] += 1
+    >>> print a['blah']
+    1
+
+    The counter also includes additional functionality useful in implementing
+    the classifiers for this assignment.  Two counters can be added,
+    subtracted or multiplied together.  See below for details.  They can
+    also be normalized and their total count and arg max can be extracted.
+    """
+    def __getitem__(self, idx):
+        self.setdefault(idx, 0)
+        return dict.__getitem__(self, idx)
+
+    def incrementAll(self, keys, count):
+        """
+        Increments all elements of keys by the same count.
+
+        >>> a = Counter()
+        >>> a.incrementAll(['one','two', 'three'], 1)
+        >>> a['one']
+        1
+        >>> a['two']
+        1
+        """
+        for key in keys:
+            self[key] += count
+
+    def argMax(self):
+        """
+        Returns the key with the highest value.
+        """
+        if len(self.keys()) == 0: return None
+        all = self.items()
+        values = [x[1] for x in all]
+        maxIndex = values.index(max(values))
+        return all[maxIndex][0]
+
+    def sortedKeys(self):
+        """
+        Returns a list of keys sorted by their values.  Keys
+        with the highest values will appear first.
+
+        >>> a = Counter()
+        >>> a['first'] = -2
+        >>> a['second'] = 4
+        >>> a['third'] = 1
+        >>> a.sortedKeys()
+        ['second', 'third', 'first']
+        """
+        sortedItems = self.items()
+        compare = lambda x, y:  sign(y[1] - x[1])
+        sortedItems.sort(cmp=compare)
+        return [x[0] for x in sortedItems]
+
+    def totalCount(self):
+        """
+        Returns the sum of counts for all keys.
+        """
+        return sum(self.values())
+
+    def normalize(self):
+        """
+        Edits the counter such that the total count of all
+        keys sums to 1.  The ratio of counts for all keys
+        will remain the same. Note that normalizing an empty
+        Counter will result in an error.
+        """
+        total = float(self.totalCount())
+        if total == 0: return
+        for key in self.keys():
+            self[key] = self[key] / total
+
+    def divideAll(self, divisor):
+        """
+        Divides all counts by divisor
+        """
+        divisor = float(divisor)
+        for key in self:
+            self[key] /= divisor
+
+    def copy(self):
+        """
+        Returns a copy of the counter
+        """
+        return Counter(dict.copy(self))
+
+    def __mul__(self, y ):
+        """
+        Multiplying two counters gives the dot product of their vectors where
+        each unique label is a vector element.
+
+        >>> a = Counter()
+        >>> b = Counter()
+        >>> a['first'] = -2
+        >>> a['second'] = 4
+        >>> b['first'] = 3
+        >>> b['second'] = 5
+        >>> a['third'] = 1.5
+        >>> a['fourth'] = 2.5
+        >>> a * b
+        14
+        """
+        sum = 0
+        x = self
+        if len(x) > len(y):
+            x,y = y,x
+        for key in x:
+            if key not in y:
+                continue
+            sum += x[key] * y[key]
+        return sum
+
+    def __radd__(self, y):
+        """
+        Adding another counter to a counter increments the current counter
+        by the values stored in the second counter.
+
+        >>> a = Counter()
+        >>> b = Counter()
+        >>> a['first'] = -2
+        >>> a['second'] = 4
+        >>> b['first'] = 3
+        >>> b['third'] = 1
+        >>> a += b
+        >>> a['first']
+        1
+        """
+        for key, value in y.items():
+            self[key] += value
+
+    def __add__( self, y ):
+        """
+        Adding two counters gives a counter with the union of all keys and
+        counts of the second added to counts of the first.
+
+        >>> a = Counter()
+        >>> b = Counter()
+        >>> a['first'] = -2
+        >>> a['second'] = 4
+        >>> b['first'] = 3
+        >>> b['third'] = 1
+        >>> (a + b)['first']
+        1
+        """
+        addend = Counter()
+        for key in self:
+            if key in y:
+                addend[key] = self[key] + y[key]
+            else:
+                addend[key] = self[key]
+        for key in y:
+            if key in self:
+                continue
+            addend[key] = y[key]
+        return addend
+
+    def __sub__( self, y ):
+        """
+        Subtracting a counter from another gives a counter with the union of all keys and
+        counts of the second subtracted from counts of the first.
+
+        >>> a = Counter()
+        >>> b = Counter()
+        >>> a['first'] = -2
+        >>> a['second'] = 4
+        >>> b['first'] = 3
+        >>> b['third'] = 1
+        >>> (a - b)['first']
+        -5
+        """
+        addend = Counter()
+        for key in self:
+            if key in y:
+                addend[key] = self[key] - y[key]
+            else:
+                addend[key] = self[key]
+        for key in y:
+            if key in self:
+                continue
+            addend[key] = -1 * y[key]
+        return addend
+
+def raiseNotDefined():
+    fileName = inspect.stack()[1][1]
+    line = inspect.stack()[1][2]
+    method = inspect.stack()[1][3]
+
+    print "*** Method not implemented: %s at line %s of %s" % (method, line, fileName)
+    sys.exit(1)
+
+def normalize(vectorOrCounter):
+    """
+    normalize a vector or counter by dividing each value by the sum of all values
+    """
+    normalizedCounter = Counter()
+    if type(vectorOrCounter) == type(normalizedCounter):
+        counter = vectorOrCounter
+        total = float(counter.totalCount())
+        if total == 0: return counter
+        for key in counter.keys():
+            value = counter[key]
+            normalizedCounter[key] = value / total
+        return normalizedCounter
+    else:
+        vector = vectorOrCounter
+        s = float(sum(vector))
+        if s == 0: return vector
+        return [el / s for el in vector]
+
+def nSample(distribution, values, n):
+    if sum(distribution) != 1:
+        distribution = normalize(distribution)
+    rand = [random.random() for i in range(n)]
+    rand.sort()
+    samples = []
+    samplePos, distPos, cdf = 0,0, distribution[0]
+    while samplePos < n:
+        if rand[samplePos] < cdf:
+            samplePos += 1
+            samples.append(values[distPos])
+        else:
+            distPos += 1
+            cdf += distribution[distPos]
+    return samples
+
+def sample(distribution, values = None):
+    if type(distribution) == Counter:
+        items = sorted(distribution.items())
+        distribution = [i[1] for i in items]
+        values = [i[0] for i in items]
+    if sum(distribution) != 1:
+        distribution = normalize(distribution)
+    choice = random.random()
+    i, total= 0, distribution[0]
+    while choice > total:
+        i += 1
+        total += distribution[i]
+    return values[i]
+
+def sampleFromCounter(ctr):
+    items = sorted(ctr.items())
+    return sample([v for k,v in items], [k for k,v in items])
+
+def getProbability(value, distribution, values):
+    """
+      Gives the probability of a value under a discrete distribution
+      defined by (distributions, values).
+    """
+    total = 0.0
+    for prob, val in zip(distribution, values):
+        if val == value:
+            total += prob
+    return total
+
+def flipCoin( p ):
+    r = random.random()
+    return r < p
+
+def chooseFromDistribution( distribution ):
+    "Takes either a counter or a list of (prob, key) pairs and samples"
+    if type(distribution) == dict or type(distribution) == Counter:
+        return sample(distribution)
+    r = random.random()
+    base = 0.0
+    for prob, element in distribution:
+        base += prob
+        if r <= base: return element
+
+def nearestPoint( pos ):
+    """
+    Finds the nearest grid point to a position (discretizes).
+    """
+    ( current_row, current_col ) = pos
+
+    grid_row = int( current_row + 0.5 )
+    grid_col = int( current_col + 0.5 )
+    return ( grid_row, grid_col )
+
+def sign( x ):
+    """
+    Returns 1 or -1 depending on the sign of x
+    """
+    if( x >= 0 ):
+        return 1
+    else:
+        return -1
+
+def arrayInvert(array):
+    """
+    Inverts a matrix stored as a list of lists.
+    """
+    result = [[] for i in array]
+    for outer in array:
+        for inner in range(len(outer)):
+            result[inner].append(outer[inner])
+    return result
+
+def matrixAsList( matrix, value = True ):
+    """
+    Turns a matrix into a list of coordinates matching the specified value
+    """
+    rows, cols = len( matrix ), len( matrix[0] )
+    cells = []
+    for row in range( rows ):
+        for col in range( cols ):
+            if matrix[row][col] == value:
+                cells.append( ( row, col ) )
+    return cells
+
+def lookup(name, namespace):
+    """
+    Get a method or class from any imported module from its name.
+    Usage: lookup(functionName, globals())
+    """
+    dots = name.count('.')
+    if dots > 0:
+        moduleName, objName = '.'.join(name.split('.')[:-1]), name.split('.')[-1]
+        module = __import__(moduleName)
+        return getattr(module, objName)
+    else:
+        modules = [obj for obj in namespace.values() if str(type(obj)) == "<type 'module'>"]
+        options = [getattr(module, name) for module in modules if name in dir(module)]
+        options += [obj[1] for obj in namespace.items() if obj[0] == name ]
+        if len(options) == 1: return options[0]
+        if len(options) > 1: raise Exception, 'Name conflict for %s'
+        raise Exception, '%s not found as a method or class' % name
+
+def pause():
+    """
+    Pauses the output stream awaiting user feedback.
+    """
+    print "<Press enter/return to continue>"
+    raw_input()
+
+
+# code to handle timeouts
+#
+# FIXME
+# NOTE: TimeoutFuncton is NOT reentrant.  Later timeouts will silently
+# disable earlier timeouts.  Could be solved by maintaining a global list
+# of active time outs.  Currently, questions which have test cases calling
+# this have all student code so wrapped.
+#
+import signal
+import time
+class TimeoutFunctionException(Exception):
+    """Exception to raise on a timeout"""
+    pass
+
+
+class TimeoutFunction:
+    def __init__(self, function, timeout):
+        self.timeout = timeout
+        self.function = function
+
+    def handle_timeout(self, signum, frame):
+        raise TimeoutFunctionException()
+
+    def __call__(self, *args, **keyArgs):
+        # If we have SIGALRM signal, use it to cause an exception if and
+        # when this function runs too long.  Otherwise check the time taken
+        # after the method has returned, and throw an exception then.
+        if hasattr(signal, 'SIGALRM'):
+            old = signal.signal(signal.SIGALRM, self.handle_timeout)
+            signal.alarm(self.timeout)
+            try:
+                result = self.function(*args, **keyArgs)
+            finally:
+                signal.signal(signal.SIGALRM, old)
+            signal.alarm(0)
+        else:
+            startTime = time.time()
+            result = self.function(*args, **keyArgs)
+            timeElapsed = time.time() - startTime
+            if timeElapsed >= self.timeout:
+                self.handle_timeout(None, None)
+        return result
+
+
+
+_ORIGINAL_STDOUT = None
+_ORIGINAL_STDERR = None
+_MUTED = False
+
+class WritableNull:
+    def write(self, string):
+        pass
+
+def mutePrint():
+    global _ORIGINAL_STDOUT, _ORIGINAL_STDERR, _MUTED
+    if _MUTED:
+        return
+    _MUTED = True
+
+    _ORIGINAL_STDOUT = sys.stdout
+    #_ORIGINAL_STDERR = sys.stderr
+    sys.stdout = WritableNull()
+    #sys.stderr = WritableNull()
+
+def unmutePrint():
+    global _ORIGINAL_STDOUT, _ORIGINAL_STDERR, _MUTED
+    if not _MUTED:
+        return
+    _MUTED = False
+
+    sys.stdout = _ORIGINAL_STDOUT
+    #sys.stderr = _ORIGINAL_STDERR
+