UniTO/anno2/Sem1/Galla/MCAD/altro/hw3sol..htm
2024-10-29 09:11:05 +01:00

367 lines
18 KiB
HTML
Executable file
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<title>Homework 3 solutions</title>
</head>
<body>
<h1>Homework 3 solutions</h1>
<ol start=1 type=1>
<li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><b>7.1.
Busy Waiting<br>
<br>
</b><span style='font-weight:normal'><i>Busy waiting</i></span> is
continuously executing (using the CPU) while waiting for something to
happen. This is typically implemented by spinning; e.g. testing a variable
in a tight loop until it changes. <br>
<br>
The alternative to busy waiting is <i>blocking</i><span style='font-style:
normal'>, where the waiting process is suspended an other processes can
execute while the process is waiting.<br style='mso-special-character:
line-break'>
<![if !supportLineBreakNewLine]><br style='mso-special-character:line-break'>
<![endif]></span></li>
<li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><b>7.2.
Spinlocks<br>
<br>
</b><span style='font-weight:normal'>Spinlocks are not appropriate for
uniprocessor systems because by definition only one process is executing
at a time. As a result, no other process can ever change release the lock
the scheduler has to intervene and switch out the spinning process
first. As a result, the spinning process might as well suspend itself
instead of spinning. Or, locking should disable interrupts to prevent the scheduler
from running, so that the process is guaranteed that it can execute the
entire critical section.<br>
<br>
On a 2 CPU multiprocessor, for example, a process scheduled on processor 1
can release the lock while a process on processor 2 is spinning on it. No
interrupt is needed. Thus, spinlocks are not necessarily wasteful on a
multiprocessor.</span></li>
</ol>
<p class=MsoNormal>Synchronization Problems</p>
<p class=MsoNormal>Notes:</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>Please use real binary semaphores, counting semaphores,
or monitors. If you invent your own synchronization method, there is a 99% chance
you will get it wrong. Plus it makes it very hard to grade - we look very
closely and are more likely to find problems. If you must do this, provide an
implementation of your construct, because otherwise we don't know if:</p>
<p class=MsoNormal style='margin-left:1.0in;text-indent:-.25in;mso-list:l2 level2 lfo7;
tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New"'>o<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>It is possible to implement</p>
<p class=MsoNormal style='margin-left:1.0in;text-indent:-.25in;mso-list:l2 level2 lfo7;
tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New"'>o<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>How it works</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>If you use semaphores, you must specify initial values
for your semaphores (mandatory).</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>If you have a shared state, you should say what mutex
protects the shared state (not mandatory)</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>With semaphores, if you wake up on one semaphore and then
call signal() to wake up the next waiter you might wake yourself up -
semaphores have history, so your next wait might be the one that awakes,
because the other waiters might have been context switchted out before they
call wait().</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>With semaphores, you shouldn't wait while holding a
mutex, unless the routine signaling you does not need the mutex to wake you up.</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>With monitors, you don't need to grab a mutex (or,
shudder, a condition variable) to access shared state. The monitor, by
definition, ensures mutual exclusion.</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>With monitors, wait() doesn't take any parameters - it
just waits until somebody then calls signal().</p>
<p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l2 level1 lfo7;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:"Times New Roman"'>-<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]>When writing synchronization code, you should try to
minimize the number of context switches and the number of times a process is
awoken when it doesn't need to be. Ideally, in a while() loop, a process should
only be woken once.<br style='mso-special-character:line-break'>
<![if !supportLineBreakNewLine]><br style='mso-special-character:line-break'>
<![endif]></p>
<ol start=3 type=1>
<li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><b>7.8.
Sleeping-barber problem.</b><span style='font-weight:normal'> </span></li>
</ol>
<h2 style='margin-left:.5in;mso-list:l0 level1 lfo10'><![if !supportLists]><span
style='font-size:12.0pt'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]><span style='font-size:12.0pt'>Barbershop Requirements:<o:p></o:p></span></h2>
<h3 style='margin-left:.5in;mso-list:l0 level1 lfo11'><![if !supportLists]><span
style='font-size:10.0pt'>-<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]><span style='font-size:10.0pt'>Barber sleeps if no
customers waiting<o:p></o:p></span></h3>
<h3 style='margin-left:.5in;mso-list:l0 level1 lfo11'><![if !supportLists]><span
style='font-size:10.0pt'>-<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]><span style='font-size:10.0pt'>Customers leave if no
chairs for waiting<o:p></o:p></span></h3>
<h3 style='margin-left:.5in;mso-list:l0 level1 lfo11'><![if !supportLists]><span
style='font-size:10.0pt'>-<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]><span style='font-size:10.0pt'>Waiting customers can't
leave until haircut done.<o:p></o:p></span></h3>
<h2 style='margin-left:.5in;mso-list:l0 level1 lfo10'><![if !supportLists]><span
style='font-size:12.0pt'><span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]><span style='font-size:12.0pt'>Solution: we use
semaphores<o:p></o:p></span></h2>
<h3 style='margin-left:.5in;mso-list:l0 level1 lfo11'><![if !supportLists]><span
style='font-size:10.0pt'>-<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]><span style='font-size:10.0pt'>Mutex = 1<o:p></o:p></span></h3>
<h3 style='margin-left:.5in;mso-list:l0 level1 lfo11'><![if !supportLists]><span
style='font-size:10.0pt'>-<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;
</span></span><![endif]><span style='font-size:10.0pt'>counting:
barber_sleeping, customer_queue, cut_done<o:p></o:p></span></h3>
<b>Barber Code</b><br>
<br>
<pre>
wait(mutex)
if (customers_waiting == 0) {
signal(mutex);
wait(barber_sleeping);
wait(mutex);
}
customers_waiting--;
signal(mutex);
signal(customer_queue);
do_cut_hair();
signal(cut_done);
</pre>
<b>Customer Code</b><br>
<pre>
wait(mutex);
if (customers_waiting == n) {
signal(mutex);
return;
}
customers_waiting++;
if (customers_waiting == 1) {
signal(barber_sleeping);
}
signal(mutex);
wait(customer_queue);
get_hair_cut();
wait(cut_done);
</pre>
<b>As a monitor</b>
<pre>
monitor barbershop {
int num_waiting;
condition get_cut;
condition barber_asleep;
condition in_chair;
condition cut_done;
Barber routine
barber() {
while (1);
while (num_waiting == 0) {
barber_asleep.wait();
}
customer_waiting.signal();
in_chair.wait();
give_hait_cut();
cut_done.signal();
}
Customer routine
customer () {
if (num_waiting == n) {
return;
}
if (num_waiting == 0) {
barber_asleep.signal();
}
customer_waiting.wait();
in_char.signal();
get_hair_cut();
cut_done.wait();
}
</pre>
<p style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></p>
<p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo3;
tab-stops:list .5in'><![if !supportLists]>7.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><![endif]><b>7.9. The Cigarette-Smokers Problem.</b><span
style='font-weight:normal'> </span></p>
<p style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Semaphores
are a convenient mechanism for implementing this, because they remember what
elements are on the table.</p>
<p style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span
style='font-family:"Courier New"'>Sempahore TobaccoAndPaper = 0;<br>
Sempahore PaperAndMatches = 0;<br>
Semaphore MatchesAndTobacco = 0;<br>
Sempaphore DoneSmoking = 1;<o:p></o:p></span></p>
<p class=MsoNormal style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span
style='font-family:"Courier New"'>void agent() <br>
{<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; wait(DoneSmoking);<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; int r = rand() % 3;<o:p></o:p></span></p>
<p class=MsoNormal style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span
style='font-family:"Courier New"'><br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; //<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; // Signal which ever combination was<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; // chosen.<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; //<br style='mso-special-character:line-break'>
<![if !supportLineBreakNewLine]><br style='mso-special-character:line-break'>
<![endif]><o:p></o:p></span></p>
<p class=MsoNormal style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span
style='font-family:"Courier New"'><br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; switch( r ) {<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; case 0:
signal(TobaccoAndPaper);<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; break;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; case 1:
signal(PaperAndMatches);<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; break;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; case 2:
signal(MatchesAndTobacco);<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; break;<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; }<br>
}<br>
<br>
void Smoker_A()<br>
{<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; while(true) {<br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; //<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; // Wait for our
two ingredients<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; //<br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
wait(TobaccoAndPaper);<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; smoke();<br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; //<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; // Signal that
we're done smoking<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; // so the next
agent can put down<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; // ingredients.<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; //<br>
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
signal(DoneSmoking);<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; }<br>
}<br style='mso-special-character:line-break'>
<![if !supportLineBreakNewLine]><br style='mso-special-character:line-break'>
<![endif]></span></p>
<p class=MsoNormal style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>Smoker
b and c are similar.<span style='font-family:"Courier New"'><o:p></o:p></span></p>
<p style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span
style="mso-spacerun: yes">&nbsp;</span></p>
<pre style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;</span></pre>
<p style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo3;
tab-stops:list .5in'><![if !supportLists]>8.<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span><![endif]><b>7.15. File-synchronization</b><span style='font-weight:
normal'> </span></p>
<p style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>
From the book you should know how to use a <i>conditional-wait</i><span
style='font-style:normal'> construct (priority-based signaling), so we'll use it here. We'll assume for simplicity that no
process has an id greater than or equal to n (or we'd just print an error
message). </span></p>
<pre style='margin-left:.5in'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></pre><pre
style='margin-left:.5in'> type file = monitor</pre><pre style='margin-left:
.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp; </span>var space_available: binary condition</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>total: integer</pre><pre
style='margin-left:.5in'> </pre><pre style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;</span>procedure entry file_open(id)</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp; </span>begin</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp; </span>while (total + id &gt;= n)</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>space_available.wait(id)</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp; </span>total = total + id;</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp; </span>if (total &lt; n - 1)</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>space_available.signal();</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp; </span>end</pre><pre
style='margin-left:.5in'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp; </span>procedure entry file_close(id)</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp; </span>begin</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp; </span>total = total - id;</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp;&nbsp; </span>space_available.signal();</pre><pre
style='margin-left:.5in'><span style="mso-spacerun: yes">&nbsp;&nbsp; </span>end</pre>
<p class=MsoNormal style='margin-left:.5in;tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'>What
happens here? As long as the incoming processes find adequate space available
(i.e. sums less than n), they will claim that space and open the file. Note
that since we're using a binary condition, we can signal() at will even if
nothing is waiting. If a process finds inadequate space, it will block. When a
process is done, it wakes up the process with the smallest id (the most likely
to fit in the available space). This process, upon waking, then signals the
next process if space is still available, but only after successfully claiming
its own space. At some point (quite possibly with the first process), the space
made available may not be enough for the waking process, and so a process will
be woken up prematurely. This is what the loop is for; it will immediately
block again. </p>
<ol start=6 type=1>
<li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><b>8.4
b<br>
</b>install traffic lights :)
<br>
<li class=MsoNormal style='mso-list:l1 level1 lfo3;tab-stops:list .5in'><b>8.8
<br>
</b>The system can always make progress in all possible
allocation scenarios (use pigeonhole principle to show this).
</ol>
<hr>
</body>
</html>