小內存堆管理算法詳細解析

1、小內存堆管理算法介紹

     本文所介紹的內存堆管理是RT Thread操作系統中的小內存管理算法,參考mem.c源文件。這個程序適用於小內存的CPU,比如像STM32F這樣的只有幾十-幾百KB內存的處理器。整個內存堆的處理算法簡潔,高效,現對其中的原理做詳細的介紹。首先先寫上整個源代碼,如下。內存堆的函數只有主要的4個函數:

  •   rt_system_heap_init
  •   rt_malloc
  •   rt_free
  •     plug_holes
/*頭文件中的相關宏定義  define in header file*/
#define HEAP_MAGIC 0x1ea0
struct heap_mem
{
    /* magic and used flag */
    rt_uint16_t magic;
    rt_uint16_t used;

    rt_size_t next, prev;

#ifdef RT_USING_MEMTRACE
    rt_uint8_t thread[4];   /* thread name */
#endif
};

/** pointer to the heap: for alignment, heap_ptr is now a pointer instead of an array */
static rt_uint8_t *heap_ptr;

/** the last entry, always unused! */
static struct heap_mem *heap_end;

/**
 * @ingroup BasicDef
 *
 * @def RT_ALIGN(size, align)
 * Return the most contiguous size aligned at specified width. RT_ALIGN(13, 4)
 * would return 16.
 */
#define RT_ALIGN(size, align)           (((size) + (align) - 1) & ~((align) - 1))

#define RT_ALIGN_SIZE         4
#define MIN_SIZE 12
#define MIN_SIZE_ALIGNED     RT_ALIGN(MIN_SIZE, RT_ALIGN_SIZE)
#define SIZEOF_STRUCT_MEM    RT_ALIGN(sizeof(struct heap_mem), RT_ALIGN_SIZE)


/*mem.c  source code below*/

static struct heap_mem *lfree;   /* pointer to the lowest free block */

static struct rt_semaphore heap_sem;
static rt_size_t mem_size_aligned;

#ifdef RT_MEM_STATS
static rt_size_t used_mem, max_mem;
#endif
#ifdef RT_USING_MEMTRACE
rt_inline void rt_mem_setname(struct heap_mem *mem, const char *name)
{
    int index;
    for (index = 0; index < sizeof(mem->thread); index ++)
    {
        if (name[index] == '\0') break;
        mem->thread[index] = name[index];
    }

    for (; index < sizeof(mem->thread); index ++)
    {
        mem->thread[index] = ' ';
    }
}
#endif

static void plug_holes(struct heap_mem *mem)
{
    struct heap_mem *nmem;
    struct heap_mem *pmem;

    RT_ASSERT((rt_uint8_t *)mem >= heap_ptr);
    RT_ASSERT((rt_uint8_t *)mem < (rt_uint8_t *)heap_end);
    RT_ASSERT(mem->used == 0);

    /* plug hole forward */
    nmem = (struct heap_mem *)&heap_ptr[mem->next];
    if (mem != nmem &&
        nmem->used == 0 &&
        (rt_uint8_t *)nmem != (rt_uint8_t *)heap_end)
    {
        /* if mem->next is unused and not end of heap_ptr,
         * combine mem and mem->next
         */
        if (lfree == nmem)
        {
            lfree = mem;
        }
        mem->next = nmem->next;
        ((struct heap_mem *)&heap_ptr[nmem->next])->prev = (rt_uint8_t *)mem - heap_ptr;
    }

    /* plug hole backward */
    pmem = (struct heap_mem *)&heap_ptr[mem->prev];
    if (pmem != mem && pmem->used == 0)
    {
        /* if mem->prev is unused, combine mem and mem->prev */
        if (lfree == mem)
        {
            lfree = pmem;
        }
        pmem->next = mem->next;
        ((struct heap_mem *)&heap_ptr[mem->next])->prev = (rt_uint8_t *)pmem - heap_ptr;
    }
}

/**
 * @ingroup SystemInit
 *
 * This function will initialize system heap memory.
 *
 * @param begin_addr the beginning address of system heap memory.
 * @param end_addr the end address of system heap memory.
 */
void rt_system_heap_init(void *begin_addr, void *end_addr)
{
    struct heap_mem *mem;
    rt_uint32_t begin_align = RT_ALIGN((rt_uint32_t)begin_addr, RT_ALIGN_SIZE);
    rt_uint32_t end_align = RT_ALIGN_DOWN((rt_uint32_t)end_addr, RT_ALIGN_SIZE);

    RT_DEBUG_NOT_IN_INTERRUPT;

    /* alignment addr */
    if ((end_align > (2 * SIZEOF_STRUCT_MEM)) &&
        ((end_align - 2 * SIZEOF_STRUCT_MEM) >= begin_align))
    {
        /* calculate the aligned memory size */
        mem_size_aligned = end_align - begin_align - 2 * SIZEOF_STRUCT_MEM;
    }
    else
    {
        rt_kprintf("mem init, error begin address 0x%x, and end address 0x%x\n",
                   (rt_uint32_t)begin_addr, (rt_uint32_t)end_addr);

        return;
    }

    /* point to begin address of heap */
    heap_ptr = (rt_uint8_t *)begin_align;

    RT_DEBUG_LOG(RT_DEBUG_MEM, ("mem init, heap begin address 0x%x, size %d\n",
                                (rt_uint32_t)heap_ptr, mem_size_aligned));

    /* initialize the start of the heap */
    mem        = (struct heap_mem *)heap_ptr;
    mem->magic = HEAP_MAGIC;
    mem->next  = mem_size_aligned + SIZEOF_STRUCT_MEM;
    mem->prev  = 0;
    mem->used  = 0;
#ifdef RT_USING_MEMTRACE
    rt_mem_setname(mem, "INIT");
#endif

    /* initialize the end of the heap */
    heap_end        = (struct heap_mem *)&heap_ptr[mem->next];
    heap_end->magic = HEAP_MAGIC;
    heap_end->used  = 1;
    heap_end->next  = mem_size_aligned + SIZEOF_STRUCT_MEM;
    heap_end->prev  = mem_size_aligned + SIZEOF_STRUCT_MEM;
#ifdef RT_USING_MEMTRACE
    rt_mem_setname(heap_end, "INIT");
#endif

    rt_sem_init(&heap_sem, "heap", 1, RT_IPC_FLAG_FIFO);

    /* initialize the lowest-free pointer to the start of the heap */
    lfree = (struct heap_mem *)heap_ptr;
}

/**
 * @addtogroup MM
 */

/**@{*/

/**
 * Allocate a block of memory with a minimum of 'size' bytes.
 *
 * @param size is the minimum size of the requested block in bytes.
 *
 * @return pointer to allocated memory or NULL if no free memory was found.
 */
void *rt_malloc(rt_size_t size)
{
    rt_size_t ptr, ptr2;
    struct heap_mem *mem, *mem2;

    if (size == 0)
        return RT_NULL;

    RT_DEBUG_NOT_IN_INTERRUPT;

    if (size != RT_ALIGN(size, RT_ALIGN_SIZE))
        RT_DEBUG_LOG(RT_DEBUG_MEM, ("malloc size %d, but align to %d\n",
                                    size, RT_ALIGN(size, RT_ALIGN_SIZE)));
    else
        RT_DEBUG_LOG(RT_DEBUG_MEM, ("malloc size %d\n", size));

    /* alignment size */
    size = RT_ALIGN(size, RT_ALIGN_SIZE);

    if (size > mem_size_aligned)
    {
        RT_DEBUG_LOG(RT_DEBUG_MEM, ("no memory\n"));

        return RT_NULL;
    }

    /* every data block must be at least MIN_SIZE_ALIGNED long */
    if (size < MIN_SIZE_ALIGNED)
        size = MIN_SIZE_ALIGNED;

    /* take memory semaphore */
    rt_sem_take(&heap_sem, RT_WAITING_FOREVER);

    for (ptr = (rt_uint8_t *)lfree - heap_ptr;
         ptr < mem_size_aligned - size;
         ptr = ((struct heap_mem *)&heap_ptr[ptr])->next)
    {
        mem = (struct heap_mem *)&heap_ptr[ptr];

        if ((!mem->used) && (mem->next - (ptr + SIZEOF_STRUCT_MEM)) >= size)
        {
            /* mem is not used and at least perfect fit is possible:
             * mem->next - (ptr + SIZEOF_STRUCT_MEM) gives us the 'user data size' of mem */

            if (mem->next - (ptr + SIZEOF_STRUCT_MEM) >=
                (size + SIZEOF_STRUCT_MEM + MIN_SIZE_ALIGNED))
            {
                /* (in addition to the above, we test if another struct heap_mem (SIZEOF_STRUCT_MEM) containing
                 * at least MIN_SIZE_ALIGNED of data also fits in the 'user data space' of 'mem')
                 * -> split large block, create empty remainder,
                 * remainder must be large enough to contain MIN_SIZE_ALIGNED data: if
                 * mem->next - (ptr + (2*SIZEOF_STRUCT_MEM)) == size,
                 * struct heap_mem would fit in but no data between mem2 and mem2->next
                 * @todo we could leave out MIN_SIZE_ALIGNED. We would create an empty
                 *       region that couldn't hold data, but when mem->next gets freed,
                 *       the 2 regions would be combined, resulting in more free memory
                 */
                ptr2 = ptr + SIZEOF_STRUCT_MEM + size;

                /* create mem2 struct */
                mem2       = (struct heap_mem *)&heap_ptr[ptr2];
                mem2->magic = HEAP_MAGIC;
                mem2->used = 0;
                mem2->next = mem->next;
                mem2->prev = ptr;
#ifdef RT_USING_MEMTRACE
                rt_mem_setname(mem2, "    ");
#endif

                /* and insert it between mem and mem->next */
                mem->next = ptr2;
                mem->used = 1;

                if (mem2->next != mem_size_aligned + SIZEOF_STRUCT_MEM)
                {
                    ((struct heap_mem *)&heap_ptr[mem2->next])->prev = ptr2;
                }
#ifdef RT_MEM_STATS
                used_mem += (size + SIZEOF_STRUCT_MEM);
                if (max_mem < used_mem)
                    max_mem = used_mem;
#endif
            }
            else
            {
                /* (a mem2 struct does no fit into the user data space of mem and mem->next will always
                 * be used at this point: if not we have 2 unused structs in a row, plug_holes should have
                 * take care of this).
                 * -> near fit or excact fit: do not split, no mem2 creation
                 * also can't move mem->next directly behind mem, since mem->next
                 * will always be used at this point!
                 */
                mem->used = 1;
#ifdef RT_MEM_STATS
                used_mem += mem->next - ((rt_uint8_t *)mem - heap_ptr);
                if (max_mem < used_mem)
                    max_mem = used_mem;
#endif
            }
            /* set memory block magic */
            mem->magic = HEAP_MAGIC;
#ifdef RT_USING_MEMTRACE
            if (rt_thread_self())
                rt_mem_setname(mem, rt_thread_self()->name);
            else
                rt_mem_setname(mem, "NONE");
#endif

            if (mem == lfree)
            {
                /* Find next free block after mem and update lowest free pointer */
                while (lfree->used && lfree != heap_end)
                    lfree = (struct heap_mem *)&heap_ptr[lfree->next];

                RT_ASSERT(((lfree == heap_end) || (!lfree->used)));
            }

            rt_sem_release(&heap_sem);
            RT_ASSERT((rt_uint32_t)mem + SIZEOF_STRUCT_MEM + size <= (rt_uint32_t)heap_end);
            RT_ASSERT((rt_uint32_t)((rt_uint8_t *)mem + SIZEOF_STRUCT_MEM) % RT_ALIGN_SIZE == 0);
            RT_ASSERT((((rt_uint32_t)mem) & (RT_ALIGN_SIZE - 1)) == 0);

            RT_DEBUG_LOG(RT_DEBUG_MEM,
                         ("allocate memory at 0x%x, size: %d\n",
                          (rt_uint32_t)((rt_uint8_t *)mem + SIZEOF_STRUCT_MEM),
                          (rt_uint32_t)(mem->next - ((rt_uint8_t *)mem - heap_ptr))));

            RT_OBJECT_HOOK_CALL(rt_malloc_hook,
                                (((void *)((rt_uint8_t *)mem + SIZEOF_STRUCT_MEM)), size));

            /* return the memory data except mem struct */
            return (rt_uint8_t *)mem + SIZEOF_STRUCT_MEM;
        }
    }

    rt_sem_release(&heap_sem);

    return RT_NULL;
}
RTM_EXPORT(rt_malloc);

2、內存堆初始化函數rt_system_heap_init

       rt_system_heap_init函數輸入2個參數,分別是內存堆的起始地址和結束地址。在RT thread操作系統中起始地址由編譯器編譯後輸出的ZI 段結束地址,即是程序使用RAM空間的最後一個地址。內存堆的結束地址就是CPU芯片的最大RAM地址,比如我使用的芯片是STM32F103RCT6,  RAM是48KB,結束地址就是0x2000C000。

       rt_system_heap_init函數對內存堆的空間進行初始化,首先對輸入的內存起始結束地址進行對齊處理,即地址要是4字節對齊。

rt_uint32_t begin_align = RT_ALIGN((rt_uint32_t)begin_addr, RT_ALIGN_SIZE);
rt_uint32_t end_align = RT_ALIGN_DOWN((rt_uint32_t)end_addr, RT_ALIGN_SIZE);

       判斷輸入的內存空間能最少容納2個內存管理結構體數據struct heap_mem。宏RT_DEBUG_NOT_IN_INTERRUPT實現關閉中斷,判斷程序目前是不是在中斷中,如果在中斷就進行輸入斷言,再打開中斷。計算出可以分配給用戶使用的內存,mem_size_aligned即整個內存堆的空間再減去2個內存管理結構的佔用的空間。

 

    RT_DEBUG_NOT_IN_INTERRUPT;

    /* alignment addr */
    if ((end_align > (2 * SIZEOF_STRUCT_MEM)) &&
        ((end_align - 2 * SIZEOF_STRUCT_MEM) >= begin_align))
    {
        /* calculate the aligned memory size */
        mem_size_aligned = end_align - begin_align - 2 * SIZEOF_STRUCT_MEM;
    }
    else
    {
        rt_kprintf("mem init, error begin address 0x%x, and end address 0x%x\n",
                   (rt_uint32_t)begin_addr, (rt_uint32_t)end_addr);

        return;
    }    

        下面是在內存堆的開始地址的12個字節做爲內存堆的頭管理結構,結束地址-12字節開始的地址做爲內存堆的尾管理結構。

/* point to begin address of heap */
    heap_ptr = (rt_uint8_t *)begin_align;

    RT_DEBUG_LOG(RT_DEBUG_MEM, ("mem init, heap begin address 0x%x, size %d\n",
                                (rt_uint32_t)heap_ptr, mem_size_aligned));

    /* initialize the start of the heap */
    mem        = (struct heap_mem *)heap_ptr;
    mem->magic = HEAP_MAGIC;
    mem->next  = mem_size_aligned + SIZEOF_STRUCT_MEM;
    mem->prev  = 0;
    mem->used  = 0;
#ifdef RT_USING_MEMTRACE
    rt_mem_setname(mem, "INIT");
#endif

    /* initialize the end of the heap */
    heap_end        = (struct heap_mem *)&heap_ptr[mem->next];
    heap_end->magic = HEAP_MAGIC;
    heap_end->used  = 1;
    heap_end->next  = mem_size_aligned + SIZEOF_STRUCT_MEM;
    heap_end->prev  = mem_size_aligned + SIZEOF_STRUCT_MEM;
#ifdef RT_USING_MEMTRACE
    rt_mem_setname(heap_end, "INIT");
#endif

    rt_sem_init(&heap_sem, "heap", 1, RT_IPC_FLAG_FIFO);

    /* initialize the lowest-free pointer to the start of the heap */
    lfree = (struct heap_mem *)heap_ptr;

      這個函數執行完成後,內存堆空間的數據結構如下圖。

 

3、內存分配函數rt_malloc

      函數開始的幾行代碼主要對申請的內存大小進行合法判斷,對申請的最小內存進行對齊處理,即能申請到的最小內存實際是12字節,即使你想申請4字節來用,內存堆中分配出來的內存實際是12字節。

     內存分配的核心算法就是for循環中的這段代碼,如下。

/*循環從當前的內存堆中找到一個內存塊,使用鏈表的操作訪問鏈表結點的下一個結點的方法進行循環*/
for (ptr = (rt_uint8_t *)lfree - heap_ptr;
         ptr < mem_size_aligned - size;
         ptr = ((struct heap_mem *)&heap_ptr[ptr])->next)
    {
        mem = (struct heap_mem *)&heap_ptr[ptr];

        if ((!mem->used) && (mem->next - (ptr + SIZEOF_STRUCT_MEM)) >= size)
        {
            /*找到一塊空的內存,並且這個內存的空間正好夠用*/
            /* mem is not used and at least perfect fit is possible:
             * mem->next - (ptr + SIZEOF_STRUCT_MEM) gives us the 'user data size' of mem */

            if (mem->next - (ptr + SIZEOF_STRUCT_MEM) >=
                (size + SIZEOF_STRUCT_MEM + MIN_SIZE_ALIGNED))
            {
                /* (in addition to the above, we test if another struct heap_mem (SIZEOF_STRUCT_MEM) containing
                 * at least MIN_SIZE_ALIGNED of data also fits in the 'user data space' of 'mem')
                 * -> split large block, create empty remainder,
                 * remainder must be large enough to contain MIN_SIZE_ALIGNED data: if
                 * mem->next - (ptr + (2*SIZEOF_STRUCT_MEM)) == size,
                 * struct heap_mem would fit in but no data between mem2 and mem2->next
                 * @todo we could leave out MIN_SIZE_ALIGNED. We would create an empty
                 *       region that couldn't hold data, but when mem->next gets freed,
                 *       the 2 regions would be combined, resulting in more free memory
                 */
                 /* 分配size+SIZEOF_STRUCT_MEM空間後,剩餘的內存堆中的空間最少還有一個能容納SIZEOF_STRUCT_MEM大小堆管理結構的空間外加最小MIN_SIZE_ALIGNED的用戶空間,這樣分配出一個內存後,剩餘的空間能形成一個帶有用戶空間的內存管理塊,能被下次分配所使用或都可以和相鄰釋放的內存進行合併*/
                ptr2 = ptr + SIZEOF_STRUCT_MEM + size;

                /* create mem2 struct */
                mem2       = (struct heap_mem *)&heap_ptr[ptr2];
                mem2->magic = HEAP_MAGIC;
                mem2->used = 0;
                mem2->next = mem->next;
                mem2->prev = ptr;
#ifdef RT_USING_MEMTRACE
                rt_mem_setname(mem2, "    ");
#endif

                /* and insert it between mem and mem->next */
                mem->next = ptr2;
                mem->used = 1;

                if (mem2->next != mem_size_aligned + SIZEOF_STRUCT_MEM)
                {
                    /*當經過free後,已用的連續內存中釋放出一個未用的內存碎片後,申請使用這個內存碎片後,會執行到這裏,即把這個內存碎片分成2片後,把後面未用的一片連接到前面已用的碎片上面*/
                    ((struct heap_mem *)&heap_ptr[mem2->next])->prev = ptr2;
                }
#ifdef RT_MEM_STATS
                used_mem += (size + SIZEOF_STRUCT_MEM);
                if (max_mem < used_mem)
                    max_mem = used_mem;
#endif
            }
            else
            {
                /* (a mem2 struct does no fit into the user data space of mem and mem->next will always
                 * be used at this point: if not we have 2 unused structs in a row, plug_holes should have
                 * take care of this).
                 * -> near fit or excact fit: do not split, no mem2 creation
                 * also can't move mem->next directly behind mem, since mem->next
                 * will always be used at this point!
                 */
                /*可用的這個內存塊分配size空間後,剩餘的空間只能放下一個內存管理塊結構後就沒有用戶空間了,對這個內存塊不再進行分塊處理了,直接使用整個塊*/
                mem->used = 1;
#ifdef RT_MEM_STATS
                used_mem += mem->next - ((rt_uint8_t *)mem - heap_ptr);
                if (max_mem < used_mem)
                    max_mem = used_mem;
#endif
            }
            /* set memory block magic */
            mem->magic = HEAP_MAGIC;
#ifdef RT_USING_MEMTRACE
            if (rt_thread_self())
                rt_mem_setname(mem, rt_thread_self()->name);
            else
                rt_mem_setname(mem, "NONE");
#endif

            if (mem == lfree)
            {
                /* Find next free block after mem and update lowest free pointer */
                while (lfree->used && lfree != heap_end)
                    lfree = (struct heap_mem *)&heap_ptr[lfree->next];

                RT_ASSERT(((lfree == heap_end) || (!lfree->used)));
            }

            rt_sem_release(&heap_sem);
            RT_ASSERT((rt_uint32_t)mem + SIZEOF_STRUCT_MEM + size <= (rt_uint32_t)heap_end);
            RT_ASSERT((rt_uint32_t)((rt_uint8_t *)mem + SIZEOF_STRUCT_MEM) % RT_ALIGN_SIZE == 0);
            RT_ASSERT((((rt_uint32_t)mem) & (RT_ALIGN_SIZE - 1)) == 0);

            RT_DEBUG_LOG(RT_DEBUG_MEM,
                         ("allocate memory at 0x%x, size: %d\n",
                          (rt_uint32_t)((rt_uint8_t *)mem + SIZEOF_STRUCT_MEM),
                          (rt_uint32_t)(mem->next - ((rt_uint8_t *)mem - heap_ptr))));

            RT_OBJECT_HOOK_CALL(rt_malloc_hook,
                                (((void *)((rt_uint8_t *)mem + SIZEOF_STRUCT_MEM)), size));

            /* return the memory data except mem struct */
            return (rt_uint8_t *)mem + SIZEOF_STRUCT_MEM;
        }
    }

    rt_sem_release(&heap_sem);

        經過連續4次內存分配並且中間不釋放內存後的內存堆分佈如下圖。此時內存是連續使用,沒有內存碎片產生。

     4、內存釋放函數rt_free

      函數前面幾行對要釋放的內存指針進行合法性檢測,判斷內存指針是否是4字節對齊,釋放的內存位置內存堆中。

       釋放一塊內存的操作很簡單,即對內存塊的使用標誌清零,mem->used=0。之後如果釋放的內存位置空閒內存塊lfree之前,那麼把空閒內存塊lfree指針指向這個剛剛釋放的內存塊。plug_holes函數實現對內存碎片進行合併處理,減少內存碎片的出現,注意並不能杜絕內存碎片的產生。

 

/* ... and is now unused. */
    mem->used  = 0;
    mem->magic = HEAP_MAGIC;
#ifdef RT_USING_MEMTRACE
    rt_mem_setname(mem, "    ");
#endif

    if (mem < lfree)
    {
        /* the newly freed struct is now the lowest */
        lfree = mem;
    }

#ifdef RT_MEM_STATS
    used_mem -= (mem->next - ((rt_uint8_t *)mem - heap_ptr));
#endif

    /* finally, see if prev or next are free also */
    plug_holes(mem);

     向一個已經連續申請的內存中間釋放一個內存塊後的內存結構分佈如下圖,釋放第三塊內存塊到內存堆中後的內存分佈圖。

5、內存合併函數 plug_holes   

      函數實現:判斷釋放的內存塊的前面一個內存塊,如果這個內存是未用的,就把這個內存與剛釋放的內存合併成一個大的內存塊。

     

/* plug hole forward */
    nmem = (struct heap_mem *)&heap_ptr[mem->next];
    if (mem != nmem &&
        nmem->used == 0 &&
        (rt_uint8_t *)nmem != (rt_uint8_t *)heap_end)
    {
        /* if mem->next is unused and not end of heap_ptr,
         * combine mem and mem->next
         */
        if (lfree == nmem)
        {
            lfree = mem;
        }
        mem->next = nmem->next;
        ((struct heap_mem *)&heap_ptr[nmem->next])->prev = (rt_uint8_t *)mem - heap_ptr;
    }

      判斷釋放的內存塊的後面的一個內存志,如果這個內存是未用的,就把這個內存與剛釋放的內存合併成一個大的內存塊,移動空閒內存指針指向前一個內存塊。通過這樣的操作就減少了內存碎片的出現,把已經釋放的相鄰的內存碎片合併成一個大的內存塊,減少內存碎片的數量。

/* plug hole backward */
    pmem = (struct heap_mem *)&heap_ptr[mem->prev];
    if (pmem != mem && pmem->used == 0)
    {
        /* if mem->prev is unused, combine mem and mem->prev */
        if (lfree == mem)
        {
            lfree = pmem;
        }
        pmem->next = mem->next;
        ((struct heap_mem *)&heap_ptr[mem->next])->prev = (rt_uint8_t *)pmem - heap_ptr;
    }

           如下圖在已經釋放了3號內存塊後,再釋放2號內存塊後,2號內存塊與3號內存塊進行合併後的內存分配圖如下。

 

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